# Properties of Matrices Transpose

Matrix Transpose

A collection of numbers arranged in the fixed number of rows and columns is called a matrix. It is a rectangular array of rows and columns. When we swap the rows into columns and columns into rows of the matrix, the resultant matrix is called the Transpose of a matrix.

This interchanging of rows and columns of the actual matrix is Matrices Transposing.

If M[ ij ] is a m x n matrix, and we want to find the transpose of this matrix, we need to interchange the rows to columns and columns to rows. It would be denoted by MT or Mâ€™. So if M = [M[ ij ] ]m x n is the original matrix, then Mâ€™ = [M[ ji ] ]n x m is the transpose of it.

For example: M =

$$\begin{array}{l}\begin{bmatrix} 2 & 3 & 4\\ 5 & 6 & 7 \end{bmatrix}\end{array}$$

the Mâ€™ =

$$\begin{array}{l}\begin{bmatrix} 2 & 5\\ 3 & 6\\ 4& 7 \end{bmatrix}\end{array}$$

In this article, let’s discuss some important properties of matrices transpose are given with example.

## Transpose Matrix Properties

Some important properties of matrices transpose are given here with the examples to solve the complex problems.

1. Transpose of transpose of a matrix is the matrix itself. [MT]T = M

For example: M =

$$\begin{array}{l}\begin{bmatrix} 2 & 3 & 4\\ 5 & 6 & 7 \end{bmatrix}\end{array}$$

the Mâ€™ =

$$\begin{array}{l}\begin{bmatrix} 2 & 5\\ 3 & 6\\ 4& 7 \end{bmatrix}\end{array}$$

and [Mâ€™]â€™ =

$$\begin{array}{l}\begin{bmatrix} 2 & 3 & 4\\ 5 & 6 & 7 \end{bmatrix}\end{array}$$

2. If thereâ€™s a scalar a, then the transpose of the matrix M times the scalar (a) is equal to the constant times the transpose of the matrix Mâ€™.Â (aM)T = aMT.

For example:

if M =

$$\begin{array}{l}\begin{bmatrix} 2 & 3 & 4\\ 5 & 6 & 7 \end{bmatrix}\end{array}$$
and constant a = 2 ,then

LHS : [aM]T = (2

$$\begin{array}{l}\begin{bmatrix} 2 & 3 & 4\\ 5 & 6 & 7 \end{bmatrix}\end{array}$$
)T

I.e

$$\begin{array}{l}\begin{bmatrix} 4 & 6 & 8\\ 10 & 12 & 14 \end{bmatrix}\end{array}$$
T

$$\begin{array}{l}\begin{bmatrix} 4 & 10\\ 6 & 12\\ 8 & 14 \end{bmatrix}\end{array}$$

RHS: a[M]T = 2 (

$$\begin{array}{l}\begin{bmatrix} 2 & 3 & 4\\ 5 & 6 & 7 \end{bmatrix}\end{array}$$
)T

= 2 (

$$\begin{array}{l}\begin{bmatrix} 2 & 5\\ 3 & 6\\ 4& 7 \end{bmatrix}\end{array}$$
)

=

$$\begin{array}{l}\begin{bmatrix} 4 & 10\\ 6 & 12\\ 8 & 14 \end{bmatrix}\end{array}$$

So, LHS = RHS

3. The sum of transposes of matrices is equal to the transpose of the sum of two

matrices.Â (M + N )T = MT + NT

M =

$$\begin{array}{l}\begin{bmatrix} 2 & 3 & 4\\ 5 & 6 & 7 \end{bmatrix}\end{array}$$

N =

$$\begin{array}{l}\begin{bmatrix} 8 & 9 & 10\\ 11 & 12 & 13 \end{bmatrix}\end{array}$$

Proof :

(M + N )T = MT + NT

LHS = (

$$\begin{array}{l}\begin{bmatrix} 2 & 3 & 4\\ 5 & 6 & 7 \end{bmatrix}+\begin{bmatrix} 8 & 9 & 10\\ 11 & 12 & 13 \end{bmatrix}\end{array}$$
)T

=

$$\begin{array}{l}(\begin{bmatrix}2 + 8 & 3 + 9 & 4 + 10\\ 5 + 11 & 6 + 12 & 7 + 13\end{bmatrix})\end{array}$$
T

=(

$$\begin{array}{l}\begin{bmatrix} 10 & 12 & 14\\ 16 & 18 & 20 \end{bmatrix}\end{array}$$
)T

=

$$\begin{array}{l}\begin{bmatrix} 10 & 16\\ 12 & 18\\ 14 & 20 \end{bmatrix}\end{array}$$

RHS =

$$\begin{array}{l}(\begin{bmatrix} 2 & 3 & 4\\ 5 & 6 & 7 \end{bmatrix})^{T} + (\begin{bmatrix} 8 & 9 & 10\\ 11 & 12 & 13 \end{bmatrix})^{T}\end{array}$$

= (

$$\begin{array}{l}\begin{bmatrix} 2 & 5\\ 3 & 6\\ 4& 7 \end{bmatrix}\end{array}$$
) +(
$$\begin{array}{l}\begin{bmatrix} 8 & 11\\ 9 & 12\\ 10 & 13 \end{bmatrix}\end{array}$$
)

= (

$$\begin{array}{l}\begin{bmatrix} 2 + 8 & 5 + 11\\ 3 + 9& 6 + 12\\ 4 + 10& 7 + 13\end{bmatrix}\end{array}$$
)

=

$$\begin{array}{l}\begin{bmatrix} 10 & 16\\ 12 & 18\\ 14 & 20 \end{bmatrix}\end{array}$$

LHS = RHS

4. The product of the transposes of two matrices in reverse order is equal to the

transpose of the product of them. (MN)T = NT MT

The above property is true for any product of any number of matrices.

LHS = (MN)T =

$$\begin{array}{l}(\begin{bmatrix} 1 & 2\\ 3 & 4\\ 5 & 6 \end{bmatrix} X \begin{bmatrix} 7 & 8\\ 9 & 10\\ 11 & 12 \end{bmatrix}) ^{T}\end{array}$$

= (

$$\begin{array}{l}\begin{bmatrix} 1 X 7 & 2 X 8\\ 3 X 9 & 4 X 10\\ 5 X 11 & 6 X 12\end{bmatrix}\end{array}$$
)T

=(

$$\begin{array}{l}\begin{bmatrix} 7 & 16\\ 27 & 40\\ 55 & 72 \end{bmatrix}\end{array}$$
)T

=

$$\begin{array}{l}\begin{bmatrix} 7 & 27 & 55\\ 16 & 40 & 72 \end{bmatrix}\end{array}$$

RHS =

$$\begin{array}{l}(\begin{bmatrix} 7 & 8\\ 9 & 10\\ 11 & 12 \end{bmatrix})^{T} X (\begin{bmatrix} 1 & 2\\ 3 & 4\\ 5 & 6 \end{bmatrix})^{T}\end{array}$$

=

$$\begin{array}{l}(\begin{bmatrix} 7 & 9 & 11\\ 8 & 10 & 12 \end{bmatrix}) \, X (\begin{bmatrix} 1 & 3 & 5\\ 2 & 4 & 6 \end{bmatrix})\end{array}$$

= (

$$\begin{array}{l}\begin{bmatrix} 7 X 1 & 9 X 3& 11 X 5\\ 8 X 2 & 10 X 4 & 12 X 6\end{bmatrix}\end{array}$$
)

= (

$$\begin{array}{l}\begin{bmatrix} 7 & 27 & 55\\ 16 & 40 & 72 \end{bmatrix}\end{array}$$
)

LHS = RHS

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