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00:08Thank you very much.
00:56Thank you very much.
01:28Thank you very much.
01:43Thank you very much.
01:50Thank you very much.
01:52Sir, today our chapter or topic is matrices.
01:55So, what is the definition of matrix and what are the types and the variations?
02:01Yes, ma'am.
02:03First, to define a matrix, I would take help of something that all of you are very much familiar with.
02:14First, let us observe a school timetable.
02:19Now, all of us are familiar with the concept of the school timetable.
02:27Slide, please.
02:29Now, if you look on the screen, the school timetable that you can see, it is a rectangular arrangement.
02:37The horizontal lines are called rows and vertical lines are called columns.
02:42On the left side, you get to see the days of the week.
02:45It is Monday, Tuesday, Wednesday, Thursday and Friday.
02:48And at the top, you get to see the periods written in a horizontal line.
02:53Now, if you note that the intersection of the first row, that is the row line related to Monday and
03:04the first column, that is the intersection of the first period, that gives us the subject that is taught on
03:11Monday in the first period.
03:12Now, the horizontal line represents all the subjects that are taught on a particular day.
03:19On the vertical lines, it represents all the subjects that are taught on the first period.
03:24Now, if we refer, say, the second column that is marked by 2 and we choose Wednesday, the intersection of
03:35the row corresponding to Wednesday and the column corresponding to 2 will give us the subject that is taught on
03:42Monday.
03:45Now, using this concept of a rectangular arrangement, let us define what is a matrix.
03:53Next slide, please.
03:57Now, see what is the definition of a matrix?
04:01A matrix is a rectangular arrangement or array of numbers, symbols, expressions arranged in rows and columns.
04:11Now, three words, it's an array of numbers, symbols and expressions.
04:17Let's see an example.
04:21Capital A is a matrix whose elements are 1, 2, 3, 4, 5 and 6.
04:28Now, clearly, these are numbers.
04:32This is a matrix whose elements are all numbers.
04:34The first horizontal line represents the first row, the second horizontal line represents the second row, the first vertical column
04:42represents the first column, the second vertical column represents the second column, and the third vertical column represents the third
04:51column over here.
04:52So, there are two rows and three columns.
04:55Now, it is very important that when we write a matrix,
04:58matrix is named using capital letters
05:01and we use either the first bracket,
05:06like over here I am writing 1, 2, 3, 4,
05:08or the third bracket, like 1, 2, 3, 4,
05:12to denote a matrix.
05:14It is a very common error.
05:15We see that students during the examination
05:18use this symbol.
05:20Now, this symbol is incorrect.
05:23This is the symbol for a determinant.
05:27Be very careful.
05:27When you are defining a matrix,
05:29then you should use the correct symbol,
05:32either this or this.
05:35The elements of a matrix are always written
05:36using small letters
05:37and the matrix itself is named using a capital letter.
05:43Now, let us see a few more examples.
05:51So, B is another matrix
05:53where the elements are x, y, z, and w.
05:56Now, these are all symbols.
05:58C is another matrix
06:02whose elements are,
06:04first element is x plus y,
06:05second element is x minus y,
06:06third x square plus y square,
06:08and fourth x square minus y square.
06:10Now, these are all mathematical expressions.
06:12So, the first matrix is using numbers,
06:16the second matrix,
06:17the elements are all symbols,
06:18the third matrix,
06:19the elements are all mathematical expressions.
06:21So, all these rectangular arrangements
06:23form a matrix.
06:25Now, the concept of matrix is clear.
06:28Let us define what is order of a matrix.
06:33Slide please.
06:34First, this last slide again.
06:38Now, order of a matrix
06:44The order of a matrix is
06:46that if a matrix A
06:48consists of
06:51m rows and n columns,
06:53then the order of a matrix is m by n.
06:57Although, we pronounce it as m by n,
06:59but when we write, write m cross n.
07:02So, if we look at the first matrix,
07:04there are two rows and three columns,
07:05so, order is two by three.
07:08The second case,
07:09there are two rows and two columns,
07:10so, order is two by two.
07:11And the third matrix,
07:13there are two rows and two columns,
07:14so, order is two by two.
07:16So, order of a matrix means
07:18first the number of rows
07:21followed by the number of columns.
07:23So, first comes the number of rows,
07:25then comes the number of columns.
07:28Now, based on this concept of order,
07:30let us define
07:31what is a comparable matrix.
07:34Next slide please.
07:37Now, comparable matrix
07:40is that
07:40if a is a matrix
07:42whose elements are
07:43one, two, three and four
07:45and b is a matrix
07:46whose elements are
07:48x, y, z and w,
07:50the two matrices
07:51having the same order
07:53are called comparable matrices.
07:56So, order of a is two by two,
07:58order of b is also two by two,
08:00so a and b are comparable matrices.
08:03Now, we define what is equal matrix.
08:06Very important.
08:07Two matrices are said to be equal
08:11if and only if they are of the same order
08:14and their corresponding elements are equal.
08:17So, let us define one more matrix C
08:20whose elements are
08:22one, two, three and four.
08:25Now, clearly, in matrix A
08:27and matrix C, we see
08:28they are of the same order.
08:30So, first condition
08:31that they must be of the same order
08:32is satisfied.
08:33The second case is that
08:34all the elements
08:36one, one, two
08:37with two, three with three
08:39four with four
08:40all the elements are equal
08:41so A and B are equal matrices.
08:44Equality of matrices is a very important concept
08:46and we will use it later today
08:49while solving a lot of questions.
08:52Let us now
08:54go to types of matrices.
08:57Now, types of matrices
08:59are defined
09:01using the number of rows
09:03and number of columns
09:04present in a matrix.
09:06So, let us
09:08see the first type
09:11that is
09:12a rectangular matrix.
09:14Now, what is a rectangular matrix?
09:15A matrix
09:16which consists of
09:17m rows and n columns
09:19where m is not equal to n
09:22is called a rectangular matrix.
09:24So, if it is a matrix
09:27elements are
09:2910, 11, 12, 13, 14, 15
09:33and clearly
09:35there are two rows
09:36and three columns.
09:40So, order
09:40is two
09:41by three
09:42because the number of rows
09:44is not equal to number of columns
09:45it is a rectangular matrix.
09:47Also, we see
09:48that
09:49the shape
09:50of such a matrix
09:51is rectangular
09:54and hence comes the name.
09:56Now, we define
09:57a row matrix.
09:59A matrix consisting of
10:01only one row of elements
10:04say 1, 2, 3
10:06is called
10:06a row matrix.
10:07So, there is only one row
10:08and
10:09there are
10:11three columns.
10:12So, if there is only one row
10:13and three columns
10:14and such a matrix
10:14is called
10:15a row matrix.
10:16the order
10:16the order of such a matrix
10:17is 1 by 3.
10:20Now, we define
10:21a column matrix.
10:24As the name suggests
10:25a column matrix
10:27is a matrix
10:27which consists
10:28of
10:29only one column.
10:31So, suppose
10:32C is a matrix
10:34the elements
10:35the elements
10:36are
10:36x, y, z
10:38there are
10:39three rows
10:40are 1
10:41are 2
10:42are 3
10:42and one column
10:44so order
10:45is 3
10:45by 1.
10:47Next type
10:48is a null matrix.
10:50Null matrix
10:50is very important
10:51because null matrix
10:53is a matrix
10:53where all the elements
10:55are 0.
10:57Now, null matrix
10:58can be of any type
10:59can be a rectangular matrix
11:00a row matrix
11:01column matrix
11:02let's see
11:04so it can be
11:05a column matrix
11:09a row matrix
11:11a square matrix
11:13that is
11:15it can be
11:15a rectangular matrix
11:16a column matrix
11:18row matrix
11:19square matrix
11:19any
11:20so if all the elements
11:21are 0
11:22then such a matrix
11:23is a null matrix
11:25or a 0 matrix
11:26Now, we define
11:28a square matrix
11:32a matrix
11:33which consists of
11:34equal number
11:35of row and column
11:36is called
11:37a square matrix
11:38in case of a square matrix
11:40as we know
11:41a square
11:41all sides are equal
11:43in a matrix
11:43if the number of rows
11:44is equal to the number of column
11:45such a matrix
11:46is a square matrix
11:48so we define it
11:49this way
11:50a11
11:52a12
11:53a21
11:54a22
11:55clearly there are
11:56two rows
11:58and two columns
12:00now it is important
12:01to note
12:01why I have used
12:02this abbreviation
12:03is that
12:03a11 is the element
12:05with the intersection
12:06of the first row
12:07and the first column
12:08a12
12:10first row
12:10second column
12:11a21
12:13second row
12:14first column
12:15and a22
12:16second row
12:17second column
12:18so the number
12:19aij
12:20i indicate the roweth position
12:22j indicates the column position
12:23now there are two rows
12:25and two columns
12:26it gives the shape
12:27of a square
12:28and it is a square matrix
12:29now in a square matrix
12:30a very important property
12:33is
12:36these elements
12:38a11
12:39a22
12:40now a11
12:41is the element
12:43with the intersection
12:43of the first row
12:44first column
12:45a22
12:45is the intersection
12:46of the second row
12:48and the second column
12:50so where i equals to j
12:53that is roweth
12:53and column position
12:54becomes equal
12:55these are called
12:56diagonal elements
12:57so a11
12:59a22
13:00are diagonal elements
13:01of a square matrix
13:02now the line
13:04passing through
13:05the diagonal element
13:06is called
13:07principal diagonal
13:09of a square matrix
13:11using this concept
13:14of principal diagonal
13:15and diagonal elements
13:16and diagonal elements
13:16we will now define
13:17what is a diagonal matrix
13:20a diagonal matrix
13:22is a matrix
13:23in which all
13:25the non-diagonal elements
13:28must be 0
13:30so if i consider
13:31a is a matrix
13:33whose elements are
13:352,000,000,1000,0004
13:40clearly
13:40other than
13:41the diagonal elements
13:42all the other elements
13:43are 0
13:44so this is an example
13:46of a diagonal matrix
13:48The last type is identity matrix.
13:52Now, in case of identity matrix, all the principal diagonal elements must be 1 and all the non-diagonal elements
14:00must be 0.
14:01So, 1, 0, 0, 0, 1, 0, 0, 0, 1.
14:10Now, this is an example of an identity matrix.
14:13Clearly, all the diagonal elements are 1 and all the other elements are 0.
14:18Now, this is an identity matrix consisting of 3 row and 3 column.
14:22So, we call it identity matrix of order 3 and we denote it as I3.
14:27If we consider identity matrix of order 2, then the elements will be 1, 0, 0, 1.
14:33This is a 2 by 2 order matrix and it's called identity matrix of order 2.
14:41So, until now, so let's have a small recap.
14:44We have learnt about what is matrix, what are the types.
14:48We have learnt about the comparable matrix and the equal matrix and up to now the identity matrix.
14:54So, there is a question which arises like, is an identity matrix also a diagonal matrix?
15:00It's a very good question, ma'am.
15:01This is a question students often feel that how to connect these two.
15:06Then, let me make it clear that in case of an identity matrix, the diagonal elements are 1 and all
15:15the non-diagonal elements are 0.
15:17So, an identity matrix clearly satisfies the definition of a diagonal matrix.
15:22And hence, we conclude that all identity matrices are diagonal matrices.
15:27But, the converse is not true.
15:30That is, if we look at A, A is a diagonal matrix, the elements are 2, 3 and 4.
15:34So, A is a diagonal matrix, but it cannot be an identity matrix.
15:40So, we conclude all identity matrices are diagonal matrices, but all diagonal matrices cannot be identity.
15:48Okay.
15:49So, may I take a call because somebody is waiting for us.
15:52Shayan Mondol is waiting with his question.
15:54Yeah.
15:55Shayan, tell us.
15:56Yeah, ma'am.
15:58Hello.
15:58Yeah.
15:59We can hear you.
16:01My question is, what is the basic difference between a set and a matrix?
16:06Okay.
16:08Okay.
16:08Now, a set is a collection of well-defined objects.
16:14Well-defined distinct objects.
16:16So, when we write a set, it consists of elements.
16:21See, this is the first four natural numbers.
16:24So, a set consists of first four natural numbers.
16:27So, it consists of elements 1, 2, 3 and 4.
16:30Now, this is, in this arrangement, we do not follow a rectangular pattern.
16:36That is, as many elements as there we write separating them by comma, there is no pattern to this arrangement.
16:44However, in case of a matrix, there is a definite pattern.
16:50That is, the elements must be written using rows and columns, in which every element has a specific position in
16:59the matrix.
17:00Whereas, in a set, if you note, if I write 1, 2, 3 and 4, and if someone else writes
17:06a 3, 2, 1, 4, it does not make much of a difference.
17:10However, if I write a matrix, 1, 2, 3 and 4, and another matrix, 1, 3, 2 and 4, these
17:18are completely different matrices.
17:20So, in matrix, the position of the element is very important.
17:24They have a different position in the matrix, but in case of a set, that is not so.
17:30Also, the arrangement, there is a specific arrangement in a matrix, but in a set, there is no such arrangement.
17:39Now, we define the operations on matrices.
17:45First operation is addition.
17:49To, say, let us consider two matrices, A and B, of the same order.
17:55Very important, all of you note, that these two matrices must be of the same order.
18:01If we take two matrices of different order, addition is not possible.
18:04So, if A and B are two matrices of the same order, then their sum, A plus B, is a
18:09matrix obtained by adding the corresponding elements of A and B.
18:15So, let us see an example.
18:18Capital A is a matrix whose elements are A1, A2, A3, A4, A5, A6.
18:33Now, A plus B, if you want to compute, first check the order, two row, three column, two row, three
18:44column.
18:45So, the orders are equal.
18:46Now, we find their sum by adding the corresponding elements.
18:51So, A1 will be added to B1.
18:53So, A1 plus B1, then A2 plus B2, A3 plus B3, A4 plus B4, A5 plus B5, A6 plus B6.
19:04So, we are adding the corresponding elements.
19:06And note that the answer will be of the same order as that of A and B.
19:11The next operation that we will define is subtraction of matrices.
19:17Now, let A and B be two matrices, again, of the same order.
19:20Also, subtraction cannot be defined if the two matrices are of different order.
19:24Then, their difference A minus B is the matrix obtained by subtracting the elements of matrix B
19:30from the corresponding elements of matrix A.
19:34So, A minus B, using the same matrices A and B, A minus B is A1 minus B1, A2 minus
19:48B2, A3 minus B3, A4 minus B4, A5 minus B5, A6 minus B6.
19:59So, we subtract all the corresponding elements A minus B.
20:04Next operation is multiplication of a matrix by a scalar.
20:10Now, if K is any scalar, what is a scalar?
20:12A scalar is any real number.
20:14If K is any scalar and A is any given matrix,
20:18then the matrix obtained by multiplying each element of A by K is called a scalar multiple of A.
20:25So, let us define the product.
20:30Using the same matrix A over here, if I choose a scalar K,
20:34then K into A is a matrix whose elements will be obtained by multiplying all these elements of A by
20:41K.
20:41So, the elements will be K, A1, K into A2, K into A3, K into A4, K into A5, K
20:50into A6.
20:53Next operation is very important.
20:55That is, multiplication of a matrix by another matrix.
21:00Now, in matrix, there are two types of multiplication.
21:02There is one is multiplication of a matrix by a scalar and second is multiplication of a matrix by a
21:06matrix.
21:09Now, in this case, let us understand the conditions that is imposed.
21:15A is a matrix and B is another matrix.
21:19The first question arises is, can we multiply any two matrices A and B?
21:23Then, if not, then what is the question?
21:26What is the condition based on which we can multiply?
21:29Now, to multiply two matrices A and B,
21:32the condition that we must check first is that
21:36the number of columns of the first matrix must be equal to the number of rows of the second matrix.
21:43That is, if A is a matrix whose order is M by B
21:47and B is a matrix whose order is P by N,
21:50then the product A, B will be defined
21:53because the number of columns of the first matrix
21:57is equal to the number of rows of the second matrix.
22:03If C is the product,
22:05then the I, Kth element of C
22:08will be obtained by the sum of the product of the elements of the Ith row of A
22:16with the corresponding elements of the Kth column of B.
22:20Let us understand this thoroughly using two examples.
22:24You can write down this question.
22:27If A is a matrix whose elements are,
22:30is a row matrix, elements are 1 and 2,
22:32and B is a matrix, elements are 3 and 4,
22:41find A, B and B into A.
22:46Then find A, B and B into A.
22:50So, let us first see order of A.
22:54There is only one row and two columns, order is 1 by 2.
22:59In B, there are two rows and one column, order is 2 by 1.
23:03But, clearly, this is 2 and this is 2.
23:06So, the number of column of A is equal to the number of rows of B.
23:10So, we can define the product AB.
23:12Then A into B
23:16will be
23:18the product of the first into the second matrix.
23:21Now, how do we multiply?
23:22The rule is we multiply row elements of the first matrix
23:27with the column elements of the second.
23:29So, 1 into 3 plus 2 into 4.
23:38So, this is 3 plus 8.
23:40That means 11.
23:42Now, clearly note that
23:46in the first matrix,
23:47the number of row is 1
23:49and in the second matrix,
23:50the number of column is 1.
23:51So, we have 1 by 1 cross 1
23:53and so, the order of the answer,
23:57that is the product AB,
23:58is also 1 by 1.
24:00Now, let us do the reverse.
24:01That is B into A.
24:03So, B is the first matrix,
24:053, 4
24:08and A is the second,
24:10which is 1 and 2.
24:11Let us check the order.
24:12Now, this is the order 2 by 1.
24:15This is the order 1 by 2.
24:16Clearly, 1
24:17and 1 over here.
24:18So, matrix multiplication is possible.
24:21Now, when we multiply,
24:22this is the first row
24:23and this is the first column.
24:25So, 3 into 1.
24:27Second row, first column,
24:294 into 1.
24:31First row, second column,
24:333 into 2.
24:34And then second row,
24:36with second column,
24:374 into 2.
24:39So, the elements are
24:413, 4, 6 and 8.
24:43Clearly, this is of order 2 by 2.
24:47It matches with this
24:48and this over here.
24:50So, we see that AB
24:53is not equal to BA.
24:55So, in case of matrices,
24:57the product AB and the product BA
24:58are not equal.
25:00So, we conclude that matrix multiplication
25:02is not commutative
25:09in general.
25:10Matrix multiplication
25:11is not commutative
25:13in just very, very important property of matrices.
25:16So, sir, here a question arises,
25:17I think,
25:18that is it possible
25:20to multiply any of the matrices?
25:22I think you have given the answer before.
25:23So, when we multiply two matrices,
25:26then we have to note
25:28that the number of the columns
25:31and the number of columns
25:33of the first matrix
25:34must be equal to number of row
25:36of the second matrix.
25:37If that is not so,
25:38then the matrix multiplication
25:39will not be possible.
25:40For example,
25:41if I consider
25:42a matrix over here,
25:441, 2 and 3
25:44and another matrix,
25:461, 2, 3 and 4.
25:48Now, if I check the order over here,
25:50there are three rows
25:51and there is one column
25:52order is 3 by 1
25:53and this is 2 by 2.
25:55Clearly, this one
25:56and over here it is 2,
25:57it doesn't match.
25:58So, matrix multiplication
26:00for these two matrices
26:01is not possible.
26:02Okay.
26:03So, now we have to take
26:05a short commercial break
26:06and after the break
26:07we will be sharply come back.
26:09Be with us.
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26:43Prism
32:19Accountible.
32:23Okay.
32:33nd
32:350 1 1 equals to 1 0 2 1 this is the equation now if I multiply this equation alpha
32:47into alpha is
32:48alpha square 0 into 1 is 0 1 into alpha is alpha plus 1 into 1 is 1 then alpha
32:55into 0 is 0 0 into
32:571 is 0 1 into 0 is 0 plus 1 into 1 is 1 so this is matrix multiplication and
33:06this is equal to 1 0
33:08to 1 now using equality of matrices we have learned if two matrices are equal then all their
33:14corresponding elements are equal so over here alpha square must be equal to 1 and alpha plus 1
33:20must be equal to 2 so alpha square equal to 1 and alpha plus 1 equal to 2 alpha square
33:28equals to 1
33:29gives us alpha as plus minus 1 and alpha plus 1 equals to 2 gives alpha equals to 1 now
33:36note we
33:36have got two different values of alpha one piece plus minus 1 another piece equals to alpha alpha
33:42equals to 1 now in the question it was I find the value within bracket s of alpha students get
33:48confused that there must be always multiple value no if there is s within bracket that can be one
33:53value or multiple value in case of matrices the answer should satisfy both the equation so alpha
34:00plus minus 1 alpha equal to 1 common value is alpha equal to 1 so alpha equals to 1 is
34:05your answer
34:06because it's a common value of the two equation one more thing if you put alpha equal to 1 over
34:11here
34:12then 1 plus 1 gives you 2 but if you put minus 1 then minus 1 plus 1 will give
34:18you 0 which doesn't
34:19match with this 2 so alpha equal to minus 1 doesn't satisfy the matrix and hence cannot be the equation
34:24so the answer here will be alpha equal to 1 now we will discuss a new topic that is transpose
34:33of a matrix
34:34now if a is a square matrix if a is any matrix of order m by n then transpose of
34:41a is a matrix of order n
34:43by m which is obtained by interchanging the rows and columns of a it is denoted by a transpose that
34:53is a
34:55with t at the top or a prime any one of the two symbols can be used now let's understand
35:02this using an
35:03example a is a matrix a1 b1 c1 a2 b2 c2 then a transpose will be obtained by interchanging the
35:22first row into the first column so a1 b1 c1 and the second row into the second column a2 b2
35:31c2 so
35:32this is a transpose change there are two row and three column over here there are three row and two
35:38columns the order gets interchanged so this is a transpose now if i say how to do this in everyday
35:46life remember the television screen that you see or the smartphone screen that you use are all pixel
35:51matrices now in this matrix when you use the option to rotate a screen then what happens is a row
35:56becomes
35:57a column and a column becomes a row so this is a practical example of use of transfers in our
36:01everyday
36:02life now let's focus on properties of the transpose yes ma'am let's focus on the properties now
36:11next slide please yes now properties of transpose the first property that you can see on the screen
36:21is a transpose whole transpose equals to a that is if i transpose a matrix row becomes column and column
36:28becomes row then again transpose it then it goes back to the original matrix a a transpose whole
36:33transpose equal to a property two a plus b whole transpose equals a transpose plus b transfer that means
36:40transpose of sum of two matrices equals to sum of their transposes third a minus b whole transpose equal to
36:48a
36:48transpose minus b transpose that means transpose of difference of two matrices is equal to difference of
36:54they are transpose. Fourth, K, where K is any scalar, that's a real number, then K into
37:00A whole transpose equals to K into A transpose. Now, understand this, that K is a scalar
37:06and so it's not affected by transpose. So, we have K into A transpose on the right side
37:10and the last very important property is A into B whole transpose is B transpose into
37:16A transpose. Now, note, the left hand side is AB whole transpose, right side is B transpose
37:23into A transpose. This property 5 is called reversal law of transposes where the product
37:30is the order in which the two matrices are multiplied is reversed after taking transpose. Now, based
37:38on the concept of transpose, we can now define two more types of matrices, that is a symmetric
37:44matrix and a skew symmetric matrix. Now, what is a symmetric matrix? If capital A U square
37:51matrix such that if A transpose equals to A, then A is called a symmetric matrix. Let
38:00us see the example. A equals to 1, 2, 2, 4. If I take transpose of this, then the first
38:10row becomes the first column 1, 2. The second row becomes the second column 2, 4 and we see
38:16there is no change to it. That is A transpose is equal to A. So, this is an example of
38:21a symmetric
38:22matrix. Now, let us consider an example of a skew symmetric matrix. A skew symmetric matrix
38:30is a square matrix in which A transpose must be equal to minus A. So, let us consider an
38:37example. A equals to 0, 5, minus 5, 0. Now, A transpose equals to, so the first row, first
38:51column becomes the first row, the second column becomes the second row. Now, we see that we
38:58have minus 5 above and 5 below. So, this is equal to minus of 0, 5, minus 5, 0 and
39:05equals
39:05to minus of A. So, clearly A transpose equal to minus of A. When A transpose equals to minus
39:12of A, then it is called a skew symmetric matrix. Now, all of you note that in the example that
39:18I selected, look at the principal diagonal elements. The principal diagonal elements of
39:24a skew symmetric matrix must always be, must always be equal to 0. This is an important
39:30characteristic of a skew symmetric matrix. So, the principal diagonal elements are all
39:350 and other elements are equal in magnitude but opposite in sign. It is as if, if you put
39:42a mirror on the principal diagonal, if you put a mirror over here, then all the elements
39:48on either side of the mirror will be equal in magnitude but only their sign will be different.
39:52Whereas, for a symmetric matrix, you put a mirror over here, then all the elements on either
40:06side will be exactly identical. That is, it will appear as if one is a reflection of the other.
40:13Now, let us focus on a few questions based on symmetric and skew symmetric matrices. Next slide,
40:21please. Okay. Before I start answering the question, let us see the result that is appearing
40:29on your screen. That is, if A is a square matrix of order N, then the following results are always
40:35true. Number one, A plus A transpose divided by 2 is a symmetric matrix. A minus A transpose
40:43divided by 2 is always a skew symmetric matrix. That is, if I add a matrix to its own transpose,
40:50then the sum divided by 2 is a symmetric and the difference divided by 2 is skew symmetric.
40:56Now, if you see that any matrix A can easily be written as A plus A transpose plus A minus
41:06A transpose. Because, if you take half common, A transpose, A transpose cancels, half into 2A
41:11is A itself. So, every square matrix A can be expressed as a sum of a symmetric matrix and
41:19a skew symmetric matrix. Now, we go to the first question. The first question says, show that
41:26A plus A transpose is symmetric if A equals to the square matrix 2, 4, 3, 5.
41:39Now, these are the elements. I have to prove A plus A transpose symmetric. Now, during the
41:43examination, take your time and do the steps correctly. The first step that you should
41:48do is calculate A transpose. That is, 2, 4, 3, 5. You do this correctly, you get your first
41:56marking. Next, we calculate A plus A transpose. That is, 2, 4, 3, 5 plus this. So, which is
42:08equal to, now if I add 2 plus 2 is 4, 4 plus 3 is 7, 3 plus 4 is
42:137, 5 plus 5 is 10. So,
42:15we get this matrix. Now, if we look, if I put a mirror over here, 7 and 7 appear as
42:21if
42:21there are images of one another. And over here, this is a symmetric matrix. Now, let's
42:26verify this. A plus A transpose whole transpose is equal to 4, 7, 7, 10 whole transpose. So,
42:36if I transpose this, I get the same matrix. That is, A plus A transpose. A plus A transpose
42:44whole transpose equals to A plus A transpose. And hence, A plus A transpose is symmetric.
42:54Okay? So, first what we did, we computed A transpose. Then, we find the sum A plus A transpose.
42:59And then, we verify that A plus A transpose whole transpose equals to A plus A transpose itself.
43:04This proves that A plus A transpose is symmetric. Now, we refer to the second question.
43:14If A equals to 5 A, B and 0. The four elements are 5 A, B and 0. And is
43:26given A is symmetric,
43:28we have to show A equal to B. Now, for this type of sum, always learn the definitions correctly.
43:33A is symmetric. This means, since A is symmetric, hence A transpose must be equal to A. Now,
43:46what is A transpose? Again, compute A transpose properly. You get your first marking for computing
43:52A transpose. So, 5 A, B, 0. A is 5 A is 0. Now, these two matrices are equal. From
44:07the
44:07concept of equality of matrices, the corresponding elements must be the same. So, if I equate
44:13these two, or if I equate these two elements, then we get A equals to B. This completes the
44:21proof. So, this type of questions are for two more questions that you get in an examination.
44:26So, computing A transpose and this one in the proof are the two important things you should
44:31focus on when you are solving this type of questions in your examination. Now, using this
44:37concept of transpose, let's see what we have covered. First, we have covered all the types
44:46of matrices, then operations, then sums on operations, then we did transpose, we defined
44:52symmetric, we defined skew symmetric matrix, then we did sums on properties of symmetric and
44:58skew symmetric matrix. Now, we have come to the last and the most important segment of this
45:03class, that is elementary operations. So, starting the elementary operation, we have
45:08someone in our phone line, Koyal Moitro is calling us. Koyal, tell us.
45:14Ma'am, my question is, what is null matrix? What is null matrix? I think sir has given the
45:20example. Yes, I have given the example. Null matrix is a matrix where all the corresponding
45:25elements are 0. That is, it can be a rectangular matrix where all the elements are 0.
45:33It can be a column matrix or it can be a row matrix. If all the elements are 0, it's
45:44a null
45:44matrix or it's also called a 0 matrix. Now, we proceed to elementary operations.
45:55Now, first, I will write four matrices on the board. When I am writing it, go through the matrices and
46:02try to find
46:03how are these matrices connected with each other. Now, A is a matrix whose elements are 1, 2, 3, 4,
46:125, 6, 7, 8, 9.
46:16B is a matrix whose elements are 4, 5, 6, 1, 2, 3, 7, 8, 9. C is a matrix
46:25whose elements are 2, 4, 6, 4, 5, 6, 7, 8, 9.
46:33And D is a matrix, 5, 7, 9, 4, 5, 6, 7, 8, 9. Observe these matrices carefully. Now, if
46:50you look at A and B, I am sure by now you have understood how are they connected.
46:54Yes, the two rows have been interchanged from the original matrix. So, what I have done over here is interchanging
47:02R1 and R2.
47:04So, the first elementary operation is interchange of two rows or two columns of a matrix. So, if I interchange
47:11two rows or two columns of a matrix, then that is an elementary operation.
47:16It is the first type operation. Now, I move to the second type. If you look at A and matrix
47:23C, what is the change?
47:26If you observe, row 2, row 3, no change. But, all the elements over here, and look at all the
47:31elements over here, 1 into 2 is 2, 2 into 2 is 4, 3 into 2 is 6.
47:37That means the operation is R1 changes to 2 into R1. All the elements of R1 have been multiplied by
47:442.
47:45That means multiplying the elements of a row by a non-zero scalar quantity K is an elementary operation. It
47:55can be a row or it can be a column.
47:58Now, A and D, if you check, then the elements of the first are obtained by adding the elements of
48:05the first over here. So, 1 plus 4 gives you 5, 2 plus 5 gives you 7, 3 plus 6
48:11gives you 9.
48:12That means over here, the operation is R1 changes to R1 plus R2. That means all the elements of R2
48:20have been actually multiplied by 1 and then added to the elements of R1.
48:25It is very important, which note that when you write this type of operation, the row that you change, that
48:32element will come first. If R1 changes, R1 will come first followed by the second row or the third row's
48:39elements you are adding to the elements of R1.
48:43So, we have discussed all the three types of elementary operation, that is the interchange of two row or column.
48:49Second is multiplying the elements of a row by a scalar quantity K, where K is not equal to 0.
48:55And third, the addition to the elements of any row or column, the corresponding elements of any other row or
49:00column multiplied by any non-zero real number, K.
49:04Now, based on this constant elementary operation, let us define equivalent matrices.
49:11Now, if you see A and B, A and C, A and D, then the matrices B, C and D
49:17has been obtained by applying some elementary operation on A.
49:20Now, then how are they related to each other?
49:23B is equivalent to A, C is equivalent to A, and D is equivalent to E.
49:29That is, if a matrix B is obtained by applying finite number of elementary operation on matrix A, then B
49:37is said to be equivalent to A.
49:39And we denote it as A, equivalent B, A, equivalent C, and A, equivalent D.
49:48This is the concept of equivalent matrices.
49:51Now, we define invertible matrices.
49:54In between them, we have a call.
49:56And Shaptam Singh is calling from Kanpur.
50:00Shaptam, we can hear you.
50:01Hello.
50:03Yes, sir.
50:04Yeah.
50:05Now, my question is, what is diagonal matrix?
50:08What is diagonal matrix?
50:10Okay, Shaptam.
50:10Sir has already described it.
50:12Yes, I have explained it when it is types of matrices.
50:15A matrix in which all the non-diagonal elements are 0 is a diagonal matrix.
50:21For example, say, 4, 0, 0, 0, 0, 5, 0, 0, 0, 7.
50:27Now, all the non-diagonal elements are 0, then such a matrix is called a diagonal matrix.
50:37Now, we define invertible matrix.
50:40A matrix A is said to be invertible if there exists a matrix B such that A into B equals
50:51to I equals to B into A.
50:53Now A is a square matrix such that there exists another matrix B if it exists then A into B
51:03equals to I equals to B into A and satisfies this condition then B is called inverse of matrix A
51:11and B is written as A inverse.
51:15Now such a matrix B may exist if it exists then A is invertible if such a matrix does not
51:23exist then A is not invertible.
51:26We have to calculate and see whether it exists or it doesn't exist.
51:30Now.
51:31So here comes I think another question that is is it possible to find the inverse of a rectangular matrix?
51:37Yes ma'am this is a very important question thank you for the question ma'am.
51:40Now inverse of a rectangular matrix students often in the moment they get a matrix start to compute the inverse
51:47but note that define that A must be a square matrix so why?
51:51The condition for invertible matrix is that the product AP must be equal to I must be equal to B
51:59into A that is A into B must be equal to B into A.
52:02Now from the concept of matrix multiplications multiplication of a matrix by a matrix right at the beginning of the
52:09class today I explained that the order is very important for matrix multiplication.
52:15The product AB and BA both multiplication is only possible if they are square matrices and at the same time
52:22because it's not only whether product is possible or not they also must be equal.
52:27So that is the corresponding elements of the product AB and the product AB must be equal this is only
52:32possible if both the matrices A and B are square matrices and not rectangular matrix so inverse of a rectangular
52:41matrix does not exist.
52:42Now using a constant element operation and invertible matrix we will now solve a question which comes for 6 months
52:50in your IS examination that is computing inverse of a matrix using elementary operations.
52:57The first question is that the first question is that a matrix is given to you whose elements are 3,
53:0310, 2, and 7.
53:08Let A equals to 3, 10, 2, 7.
53:14This is the matrix.
53:15Now if you are using elementary operation remember 3 things.
53:19Point 1.
53:20You must apply same operations on both sides.
53:24The operation that you are applying on the left should be also applied on the right.
53:27Second, when you are doing elementary operation either apply elementary row operation throughout the sum or apply elementary column operation
53:35throughout the sum.
53:36Do not mix them up.
53:37We often say students have applied two steps.
53:39They apply row operation then they start with column operation.
53:41That is not allowed and that is not permissible.
53:43So either apply row or apply column operations.
53:46Apply both the operations on both sides.
53:49If you are using row operation then you will write A equals to I A and if you are using
53:57column operation then you will write A equals to A I.
54:01Now after applying writing this we write down the matrix 3, 10, 2, 7, 1, 0, 0, 0, 1, 0,
54:110, 1 into A.
54:14Now the whole objective is that to obtain a matrix B over here such that the product will be equal
54:21to identity.
54:22So we want to convert this left hand side matrix into identity.
54:25What is identity?
54:261, 0, 0, 1.
54:28So the main objective is to bring 1 over here and 0 below.
54:31So the first operation to get 1 over here and applying R1 changes R1 minus R2.
54:39So 3 minus 2, 1, 10 minus 7 is 3.
54:43The second row remains as it is.
54:46Now don't forget if you do any operation on the left side the same operation will be on the right
54:51side.
54:52So 1 minus 0, 1, 0 minus 1 is minus 1, 0, 1.
54:56So this is the first step.
54:58We obtained 1.
54:59Now using this 1 that we have obtained we want to get 0 over here.
55:03How to do it?
55:04If I multiply this 1 by 2 and subtract then 2 minus 2 into 1 will give me 0.
55:10So my second operation is R2 changes R2 minus 2 R1.
55:16Okay.
55:16So the first row remains as it is.
55:192 minus 2 into 1, 0.
55:21So 7 minus 2 into 3, 2 into 3 is 6, 7 minus 6 is 1.
55:27So 1 minus 1, 0 minus 2 into 1 is minus 2, 1 minus 2 into minus 1 that is
55:36plus 2.
55:37So 1 plus 2 is 3.
55:39So now it's time to give the assignments to the students.
55:42Okay.
55:43Now just one step left.
55:45Using this one we get 0 over here.
55:46So the operation will be R1 changes R1 minus 3, R2.
55:53So 1, 0, 0, 1.
55:58So 1 minus 3 into 2 is over here 6 is plus 1 is 7.
56:04Now minus 3 into, I say is minus 9, minus 9, minus 1 will be minus 10, minus 2, 3.
56:14So clearly I equals to B into A, therefore P equals to A inverse.
56:21That is 7 minus 10, minus 2 and 3 is the EB inverse of matrix A.
56:30This is how applying elementary operation we can find inverse of a matrix.
56:34There are two other questions given over here.
56:36Now remember, while finding inverse of a matrix, say you get 2, 0 in a row or 2, 0 in
56:41a column,
56:42then inverse of such a matrix does not exist.
56:45Okay, so please, now it's time to give the assignment.
56:48Sure ma'am.
56:48Now we move to the assignment.
56:53The first question is find a matrix A such that 2A plus 3X equals to 5B where A and B
56:57are given.
56:59Second question, if A equals to 1, 2, 2, 1, then show that A squared minus 3A equals to 2
57:04into A.
57:06Third question, if A equals to 3, 1, 7, 5, find the values of X and Y such that A
57:11squared plus X into I2 equals to 1 into A.
57:14And I2 is identity matrix of order 2.
57:16And finally, two questions of inverse are given.
57:19Now we have to make a note that when I am getting 1, 0, 0, when it is using 3
57:25cross 3 order matrix,
57:26then getting 1, 0, 0, you get your first marking.
57:30Then when you obtain,
57:34and if you obtain this by applying elementary operation, you get your first marking.
57:38Then when you apply elementary operation on the second row, you obtain 0, 1, 0, you get your second marking.
57:44And then finally, when you obtain elementary operation on the third row, you get 0, 0, 1.
57:49And at the same time, writing inverse correctly, you get your final mark.
57:53I think there should not be much questions and doubts to the students.
57:58You have such descriptively answered all the questions.
58:03And now it's time to conclude the program.
58:06And tomorrow there will be the ICC Maths program from 12 to 1 p.m.
58:14So, I think there should not be much questions.
58:17Those are going to be the traffic.
58:18And today, one guy will find the Katherine denominator and the Frankly!..
58:24He has read close number 1 of its students and the proper teachers.
58:26Now, the Eagles will compare the English and the great teachers from 12 to 3.
58:30And the них will touch as they are true.
58:32Now, I think there will be all the people in the classroom.
58:34Now, welcome to thenicos Digital Seven.
58:36Bye-bye, Maradie.
58:36Bye-bye.
58:38Bye-bye.
58:39Bye-bye.
58:40Bye-bye.
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