## How do you Transpose a Matrix?

### Transpose of a matrix

The matrix obtained from a given matrix A by changing its rows into columns or columns into rows is called transpose of matrix A and is denoted by A^{T} or A′.

From the definition it is obvious that if order of A is m × n, then order of A^{T} is n × m.

**Example:**

### Properties of transpose

Let *A* and *B* be two matrices then,

- (A
^{T})^{T}= A - (A + B)
^{T}= A^{T}+ B^{T}, A and*B*being of the same order - (kA)
^{T}= kA^{T}, k be any scalar (real or complex) - (AB)
^{T}= B^{T}A^{T}, A and*B*being conformable for the product*AB* - (A
_{1 }A_{2}A_{3}…… A_{n-1}A_{n})^{T}= A_{n}^{T}A_{n-1}^{T}……. A_{3}^{T}A_{2}^{T}A_{1}^{T} - I
^{T}= I

### Symmetric and Skew-symmetric Matrices

**(1) Symmetric matrix :**

A square matrix A = [a_{ij}] is called symmetric matrix if a_{ij} = a_{ji} for all *i*, *j* or A^{T} = A.

**(2) Skew-symmetric matrix :**

A square matrix A = [a_{ij}] is called skew- symmetric matrix if a_{ij} = −a_{ji} for all *i, j* or A^{T} = −A.

All principal diagonal elements of a skew- symmetric matrix are always zero because for any diagonal element.

a_{ij} = −a_{ji }⇒ a_{ij} = 0

### Properties of symmetric and skew-symmetric matrices

- If
*A*is a square matrix, then A + A^{T}, AA^{T}, A^{T}A are symmetric matrices, while A − A^{T}is skew- symmetric matrix. - If
*A*is a symmetric matrix, then −A, KA, A^{T}, A^{n}, A^{−1}, B^{T}AB are also symmetric matrices, where n ∈ N, K ∈ R and*B*is a square matrix of order that of*A*. - If A is a skew-symmetric matrix, then

(a) A^{2n}is a symmetric matrix for n ∈ N.

(b) A^{2n+1}is a skew-symmetric matrix for n ∈ N.

(c)*kA*is also skew-symmetric matrix, where k ∈ R.

(d) B^{T}AB is also skew- symmetric matrix where*B*is a square matrix of order that of*A.* - If
*A*,*B*are two symmetric matrices, then

(a) A ± B, AB + BA are also symmetric matrices,

(b) AB – BA is a skew- symmetric matrix,

(c) AB - If
*A*,*B*are two skew-symmetric matrices, then

(a) A ± B, AB – BA are skew-symmetric matrices,

(b) AB + BA is a symmetric matrix. - If
*A*a skew-symmetric matrix and*C*is a column matrix, then C^{T}AC is a zero matrix.