# What is Transpose of Matrix – Definition and Example

Here you will learn what is transpose of matrix with definition and examples.

Let’s begin –

## What is Transpose of Matrix

Let A = $$[a_{ij}]$$ be a $$m\times n$$ matrix. Then the transpose of A, denoted by $$A^{T}$$ or A’, is an $$n\times m$$ matrix such that

$$(A^T)_{ij}$$ = $$a_{ji}$$ for all i = 1, 2, ….. , m;  j = 1, 2, ….., n.

Thus, $$A^{T}$$ is obtained from A by changing its rows into columns and columns into rows.

for example, if A = $$\begin{bmatrix} 1 & 2 & 3 & 4\\ 2 & 3 & 4 & 1\\ 3 & 2 & 1 & 4 \end{bmatrix}$$,

then $$A^T$$ = $$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 2 \\ 3 & 4 & 1 \\ 4 & 1 & 4 \end{bmatrix}$$

The first row of $$A^T$$  is the first column of A. The second row of $$A^T$$ is the second column of A and so on.

## Properties of Transpose

(a) for any matrix A, $$(A^T)^T$$ = A.

(b) for any two matrices A and B of the same order, $$(A + B)^T)$$ = $$A^T$$ + $$B^T$$.

(c) If A is a matrix and k is a scalar, then $$(kA)^T$$ = k$$(A^T)$$.

(d) If A and B are two matrices such that AB is defined, then $$(AB)^T$$ = $$B^T$$$$A^T$$.

Generalisation : If A, B, C are three matrices confirmable for the products (AB)C and A(BC), then $$(ABC)^T$$ = $$C^T$$$$B^T$$$$A^T$$.

The above law is called reversal law for transposes i.e. the transpose of the product is the product of the transposes taken in the reverse order.

Example : If A = $$\begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}$$ and B = $$\begin{bmatrix} -2 & -1 & -4 \end{bmatrix}$$, verify that $$(AB)^T$$ = $$B^T$$ $$A^T$$

Solution : We have,

A = $$\begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}$$ and B = $$\begin{bmatrix} -2 & -1 & -4 \end{bmatrix}$$

$$\therefore$$ AB = $$\begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}$$ $$\begin{bmatrix} -2 & -1 & -4 \end{bmatrix}$$ = $$\begin{bmatrix} 2 & 1 & 4 \\ -4 & -2 & -8 \\ -6 & -3 & -12 \end{bmatrix}$$

$$\implies$$ $$(AB)^T$$ = $$\begin{bmatrix} 2 & -4 & -6 \\ 1 & -2 & -3 \\ 4 & -8 & -12 \end{bmatrix}$$

Also, $$B^T$$ = $$\begin{bmatrix} -2 \\ -1 \\ -4 \end{bmatrix}$$ and $$A^T$$ = $$\begin{bmatrix} -1 & 2 & 3 \end{bmatrix}$$

$$B^T$$$$A^T$$ = $$\begin{bmatrix} -2 \\ -1 \\ -4 \end{bmatrix}$$ $$\begin{bmatrix} -1 & 2 & 3 \end{bmatrix}$$ = $$\begin{bmatrix} 2 & 1 & 4 \\ -4 & -2 & -8 \\ -6 & -3 & -12 \end{bmatrix}$$

Hence $$(AB)^T$$ = $$B^T$$ $$A^T$$