`color {red} **` Transpose of a Matrix

`color {red} **` Properties of transpose of the matrices

`color {red} **` Symmetric and Skew Symmetric Matrices

`color {red} **` Properties of transpose of the matrices

`color {red} **` Symmetric and Skew Symmetric Matrices

`\color{green} ✍️` If `A = [a_(ij)]` be an `m × n` matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of `A. `

`\color{green} ✍️` Transpose of the matrix A is denoted by A′ or `(A^T)`. In other words, if `A = [a_(ij)]_(m × n)`, then `A′ = A^T= [a_(ji)]_(n × m)`.

● For example, If `A = [(3,5),(sqrt3,1),(0,{-1}/5) ] _(3xx2)`, then `A' = [ (3, sqrt3, 0 ),(5,1 ,{-1}/5) ]_(2xx3)`

`\color{green} ✍️` Transpose of the matrix A is denoted by A′ or `(A^T)`. In other words, if `A = [a_(ij)]_(m × n)`, then `A′ = A^T= [a_(ji)]_(n × m)`.

● For example, If `A = [(3,5),(sqrt3,1),(0,{-1}/5) ] _(3xx2)`, then `A' = [ (3, sqrt3, 0 ),(5,1 ,{-1}/5) ]_(2xx3)`

● For any matrices A and B of suitable orders, and transpose are `A' & B',` then we have

(i) (A′)′ = A,

(ii) (kA)′ = kA′ (where k is any constant)

(iii) (A + B)′ = A′ + B′

(iv) (A B)′ = B′ A′

(i) (A′)′ = A,

(ii) (kA)′ = kA′ (where k is any constant)

(iii) (A + B)′ = A′ + B′

(iv) (A B)′ = B′ A′

`color{blue}{"Symmetric Matrix :"}`

`\color{green} ✍️` A square matrix `A = [a_(ij)]` is said to be symmetric

`color{red}{"If "A′ = A, \ \ "that is,"\ \ [a_(ij)] = [a_(ji)]}` for all possible values of i and j.

For example `A = [(sqrt3, 2, 3),(2, -1.5, -1),(3, -1,1)]` is a symmetric matrix as A′ = A

`color{blue}{"Skew Symmetric Matrix :"}`

`\color{green} ✍️` A square matrix A = [aij] is said to be skew symmetric matrix

`color{red}{"if" A′ = – A,\ \ "that is" \ \ a_(ji) = – a_(ij)` for all possible values of `i` and `j.`

Now, if we put `i = j, `we have `a_(ii) = – a_(ii)`. Therefore `2a_(ii) = 0` or `a_(ii )= 0` for all `i’`s.

`=> color{green}{"This means that all the diagonal elements of a skew symmetric matrix are zero."}`

For example, the matrix `B = [ (0,e,f),(-e ,0 ,g),(-f , -g , 0) ]` is a skew symmetric matrix as B′= –B

Now, we are going to prove some results of symmetric and skew-symmetric matrices.

`\color{green} ✍️` A square matrix `A = [a_(ij)]` is said to be symmetric

`color{red}{"If "A′ = A, \ \ "that is,"\ \ [a_(ij)] = [a_(ji)]}` for all possible values of i and j.

For example `A = [(sqrt3, 2, 3),(2, -1.5, -1),(3, -1,1)]` is a symmetric matrix as A′ = A

`color{blue}{"Skew Symmetric Matrix :"}`

`\color{green} ✍️` A square matrix A = [aij] is said to be skew symmetric matrix

`color{red}{"if" A′ = – A,\ \ "that is" \ \ a_(ji) = – a_(ij)` for all possible values of `i` and `j.`

Now, if we put `i = j, `we have `a_(ii) = – a_(ii)`. Therefore `2a_(ii) = 0` or `a_(ii )= 0` for all `i’`s.

`=> color{green}{"This means that all the diagonal elements of a skew symmetric matrix are zero."}`

For example, the matrix `B = [ (0,e,f),(-e ,0 ,g),(-f , -g , 0) ]` is a skew symmetric matrix as B′= –B

Now, we are going to prove some results of symmetric and skew-symmetric matrices.

`color{blue}{" For any square matrix A with real number entries, A + A′ is a symmetric matrix"}`

`color{blue}{" and A – A′ is a skew symmetric matrix."}`

● Let B = A + A′, then

B′ = (A + A′)′

= A′ + (A′)′ (as (A + B)′ = A′ + B′)

= A′ + A (as (A′)′ = A)

= A + A′ (as A + B = B + A)

= B

● Therefore B = A + A′ is a symmetric matrix

Now let C = A – A′

C′ = (A – A′)′ = A′ – (A′)′

= A′ – A

= – (A – A′) = – C

Therefore C = A – A′ is a skew symmetric matrix.

`color{blue}{" and A – A′ is a skew symmetric matrix."}`

● Let B = A + A′, then

B′ = (A + A′)′

= A′ + (A′)′ (as (A + B)′ = A′ + B′)

= A′ + A (as (A′)′ = A)

= A + A′ (as A + B = B + A)

= B

● Therefore B = A + A′ is a symmetric matrix

Now let C = A – A′

C′ = (A – A′)′ = A′ – (A′)′

= A′ – A

= – (A – A′) = – C

Therefore C = A – A′ is a skew symmetric matrix.

`color{blue}{"Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix."}`

● Let A be a square matrix, then we can write

`A = 1/2(A +A') + 1/2 (A - A')`

● From the Theorem 1, we know that (A + A′) is a symmetric matrix and (A – A′) is a skew symmetric matrix.

● Since for any matrix A, (kA)′ = kA′, it follows that `1/2(A +A')` is symmetric matrix and `1/2 (A -A')` is skew symmetric matrix.

● Thus, any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

● Let A be a square matrix, then we can write

`A = 1/2(A +A') + 1/2 (A - A')`

● From the Theorem 1, we know that (A + A′) is a symmetric matrix and (A – A′) is a skew symmetric matrix.

● Since for any matrix A, (kA)′ = kA′, it follows that `1/2(A +A')` is symmetric matrix and `1/2 (A -A')` is skew symmetric matrix.

● Thus, any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

Q 3114680550

Express the matrix `B= [ (2,-2,-4), ( -1,3,4), ( 1,-2,-3) ] ` as the sum of a symmetric and a skew symmetric matrix.

Class 12 Chapter 3 Example 22

Class 12 Chapter 3 Example 22

Here

`B' = [ (2,-1,1), (-2,3,-2), ( -4,4,-3) ]`

Let ` P= 1/2 (B+B') =1/2 [ ( 4, -3,-3), ( -3,6,2), ( -3,2, -6) ] = [ (2, -3/2, -3/2), ( -3/2 ,3,1), (-3/2,1,-3) ]`

Now `P' = [ ( 2, -3/2 , -3/2 ), ( -3/2 ,3,1), ( -3/2 ,1, -3) ] =P`

Thus `P= 1/2 (B+B') ` is a symmetric matrix

Also, let `Q =1/2 (b -B') =1/2 [ (0,-1, -5), ( 1,0, 6), ( 5, -6 , 0 ) ] = [ (0 , -1/2, -5/2 ), ( 1/2, 0 ,3 ), ( 5/2 , -3, 0 ) ]`

Then `Q' = [ (0,1/2 , 5/3 ), (-1/2 , 0, -3 ), ( -5/2 ,3, 0 ) ] = -Q`

Thus `Q = 1/2 (B-B') `is a skew symmetric matrix.

Now `P+Q = [ (2, -3/2, -3/2 ), ( -3/2 ,3 ,1 ), ( -3/2 , 1, -3 ) ] + [ ( 0 , -1/2 , -5/2 ), ( 1/2 , 0 ,3 ), (5/2 , -3 , 0) ] = [ ( 2,-2, -4 ), (-1,3,4), ( 1,-2,-3) ] = B`

Thus, B is represented as the sum of a symmetric and a skew symmetric matrix.