`color {red} **` Transpose of a Matrix
`color {red} **` Properties of transpose of the matrices
`color {red} **` Symmetric and Skew Symmetric Matrices
`color {red} **` Transpose of a Matrix
`color {red} **` Properties of transpose of the matrices
`color {red} **` Symmetric and Skew Symmetric Matrices
Transpose of a Matrix
`\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} ✍️` 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)`
Properties of transpose of the matrices
● 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′
● 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′
Symmetric and Skew Symmetric Matrices
`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{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.
Theorem 1 :
`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}{" 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.
Theorem 2 :
`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.
`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.
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