# How do you Transpose a Matrix?

## 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 AT or A′.
From the definition it is obvious that if order of A is m × n, then order of AT is n × m.
Example:

### Properties of transpose

Let A and B be two matrices then,

1. (AT)T = A
2. (A + B)T = AT + BT, A and B being of the same order
3. (kA)T = kAT, k be any scalar (real or complex)
4. (AB)T = BTAT, A and B being conformable for the product AB
5. (A1 A2 A3 …… An-1 An)T = AnT An-1T ……. A3T A2T A1T
6. IT = I

### Symmetric and Skew-symmetric Matrices

(1) Symmetric matrix :
A square matrix A = [aij] is called symmetric matrix if aij = aji for all i, j or AT = A.

(2) Skew-symmetric matrix :
A square matrix A = [aij] is called skew- symmetric matrix if aij = −aji for all i, j or AT = −A.

All principal diagonal elements of a skew- symmetric matrix are always zero because for any diagonal element.
aij = −aji ⇒ aij = 0

### Properties of symmetric and skew-symmetric matrices

1. If A is a square matrix, then A + AT, AAT, ATA are symmetric matrices, while A − AT is skew- symmetric matrix.
2. If A is a symmetric matrix, then −A, KA, AT, An, A−1, BT AB are also symmetric matrices, where n ∈ N, K ∈ R and B is a square matrix of order that of A.
3. If A is a skew-symmetric matrix, then
(a) A2n is a symmetric matrix for n ∈ N.
(b) A2n+1 is a skew-symmetric matrix for n ∈ N.
(c) kA is also skew-symmetric matrix, where k ∈ R.
(d) BT AB is also skew- symmetric matrix where B is a square matrix of order that of A.
4. 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 is a symmetric matrix, when AB = BA.
5. 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.
6. If A a skew-symmetric matrix and C is a column matrix, then CT AC is a zero matrix.