# Transpose of a Matrix

## Definition of the Transpose of a Matrix

If M is an m × n matrix, then the transpose of M , denoted by MT, is the n × m matrix obtained by interchanging the rows and columns of matrix M .
These are examples of the transpose of matrices.
a)

Matrix A has one row and a size (or order) 1 × 3 . The transpose of matrix A is obtained by interchanging the row of the matrix into a column. Hence the transpose of matrix A has a size 3 × 1 and denoted by AT is given by

b)

The transpose of matrix B , which has one column and a size 4 × 1 , is obtained by interchanging the column of the matrix into a row. Hence the transpose of matrix B has an order 1 × 4 and denoted by BT is given by

c)

Matrix C has a size 2 × 3 . The transpose of this matrix is obtained by interchanging the rows of the matrix into columns (or columns into rows). Hence the transpose CT of matrix C has an order 3 × 2 and is given by

d)
$$D = \begin{bmatrix} 2 & - 5 & 9 \\ -7 & 0 & 9 \\ 1 & -2 & 11 \end{bmatrix}$$
The transpose of matrix $$D$$ with order $$3 \times 3$$ is obtained by interchanging the rows of the matrix into columns (or columns into rows). Hence the transpose $$D^T$$ of matrix $$D$$ has an order $$3 \times 3$$ and is given by
$$D^T = \begin{bmatrix} 2 & -7 & 1 \\ - 5 & 0 & -2 \\ 9 & 9 & 11 \end{bmatrix}$$
Note that the rows the transpose of a given matrix are the columns of the matrix and the columns of the transpose are the rows of the matrix.

## Properties of the Transpose of Matrices

Some of the most important properties of the transpose of matrices are given below.

1.   $$(A^T)^T = A$$.
2.   $$(AB)^T = B^T A^T$$
3.   $$(A+B)^T = A^T + B^T$$
4.   $$(k A)^T = k A^T$$ , k is a real number.
5.   $$(A^T)^{-1} = (A^{-1})^T$$
6.   $$Det(A^T) = Det(A)$$.
7.   $$A^T = A$$ if and only if $$A$$ is a symmetric matrix.
8.   $$A^{-1} = A^T$$ if and only if $$A$$ is an orthogonal (square) matrix.

## Examples with Solutions

Example 1
Find the transpose of the matrices:
a) $$A = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}$$      b) $$B = \begin{bmatrix} -2 & 0 & 1 \\ \end{bmatrix}$$      c) $$C = \begin{bmatrix} 5 & -7 \\ 1 & -4 \\ 0 & -1 \\ 7 & -4 \\ \end{bmatrix}$$

Solution
We find the transpose of a matrix by interchanging the rows and columns as follows:
a) $$A^T = \begin{bmatrix} 0 & 1\\ 0 & 0 \\ 1 & 0 \end{bmatrix}$$      b) $$B^T = \begin{bmatrix} -2 \\ 0 \\ 1 \end{bmatrix}$$      c) $$C^T = \begin{bmatrix} 5 & 1 & 0 & 7 \\ -7 & -4 & -1 & -4 \end{bmatrix}$$

Example 2
Matrices $$A$$ and $$B$$ are given by $$A = \begin{bmatrix} -1 & 0 & 1 \\ 1 & 2 & 0 \end{bmatrix}$$ , $$B = \begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}$$.
Show that $$(AB)^T = B^T A^T$$ (verify property 2 above).

Solution
Calculate $$AB$$
$$AB = \begin{bmatrix} -1 & 0 & 1 \\ 1 & 2 & 0 \end{bmatrix} \begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$$
Determine $$(AB)^T$$
$$(AB)^T = \begin{bmatrix} 3 & 1 \end{bmatrix}$$

Determine $$A^T$$ and $$B^T$$
$$A^T = \begin{bmatrix} -1 & 1 \\ 0 & 2 \\ 1 & 0 \end{bmatrix}$$ , $$B^T = \begin{bmatrix} -1 & 1 & 2 \end{bmatrix}$$
Calculate $$B^T A^T$$
$$B^T A^T = \begin{bmatrix} -1 & 1 & 2 \end{bmatrix} \begin{bmatrix} -1 & 1 \\ 0 & 2 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 3 & 1 \end{bmatrix}$$
Hence $$(AB)^T = B^T A^T$$

Example 3
Let matrix $$A = \begin{bmatrix} -1 & 0\\ 1 & 2 \end{bmatrix}$$
Show that $$(A^T)^{-1} = (A^{-1})^T$$ (verify property 5 above).

Solution
Determine $$A^T$$
$$A^T = \begin{bmatrix} -1 & 1\\ 0 & 2 \end{bmatrix}$$
Use the formula of the inverse of a $$2 \times 2$$ matrix $$\begin{bmatrix} x & y \\ z & w \\ \end{bmatrix}^{-1} = \dfrac{1}{xw - yz} \begin{bmatrix} w & - y \\ - z & x \\ \end{bmatrix}$$ to find $$(A^T)^{-1}$$
$$(A^T)^{-1} = -\dfrac{1}{2} \begin{bmatrix} 2 & 0\\ 1 & -1 \end{bmatrix}$$

$$\quad \quad = \begin{bmatrix} -1 & 0\\ -\dfrac{1}{2} & \dfrac{1}{2} \end{bmatrix}$$

Use the same formula of the inverse of a $$2 \times 2$$ matrix given above to find $$A^{-1}$$
$$A^{-1} = -\dfrac{1}{2} \begin{bmatrix} 2 & 1\\ 0 & -1 \end{bmatrix}$$

$$\quad \quad = \begin{bmatrix} -1 & -\dfrac{1}{2}\\ 0 & \dfrac{1}{2} \end{bmatrix}$$

We now determine $$(A^{-1}) ^T$$
$$(A^{-1}) ^T = \begin{bmatrix} -1 & 0 \\ -\dfrac{1}{2} & \dfrac{1}{2} \end{bmatrix}$$

Hence we conclude that $$(A^T)^{-1} = (A^{-1})^T$$.

Example 4
Let matrix $$A = \begin{bmatrix} -1 & 0 & 1\\ 1 & 2 & 2 \\ -2 & 0 & 1 \end{bmatrix}$$
Show that $$Det(A^T) = Det(A)$$ (verify property 6 above)

Solution
Calculate $$Det(A)$$ using the upper row
Calculate $$Det(A) = -1 ( 2 \times 1 - 0) + 1 (0 - 2(-2) ) = 2$$

Determine $$A^T$$
$$A^T \begin{bmatrix} -1 & 1 & -2 \\ 0 & 2 & 0 \\ 1 & 2 & 1 \end{bmatrix}$$
Calculate $$Det(A^T)$$ using the leftmost column
$$Det(A^T) = -1 ( 2 \times 1 - 0) + 1 ( 0 - 2(-2)) = 2$$
Hence we conclude that $$Det(A^T) = Det(A)$$

Example 5
Use property 7 above to show that matrix $$A = \begin{bmatrix} 0 & -1 & 3\\ -1 & 2 & 5 \\ 3 & -5 & 1 \end{bmatrix}$$ is not a symmetric matrix.

Solution
Determine $$A^T$$
$$A^T = \begin{bmatrix} 0 & -1 & 3 \\ -1 & 2 & -5 \\ 3 & 5 & 1 \end{bmatrix}$$
We can verify that the entry at $$A_{2,3} = 5$$ and the entry at $$A^T_{2,3} = - 5$$, therefore matrices $$A^T$$ and $$A$$ are not equal which means that matrix $$A$$ is not symmetric.

## Questions (with solutions given below)

• Part 1
Given matrix $$A = \begin{bmatrix} 1 & 1 & -3 \\ -1 & 0 & 0 \end{bmatrix}$$, calculate and show that the matrices $$A^T A$$ and $$A A^T$$ are both symmetric.
• Part 2
Given the matrices $$A = \begin{bmatrix} -5 & -2 \\ - 1 & 2 \end{bmatrix}$$ and $$B = \begin{bmatrix} 0 & 3 \\ 2 & -3 \end{bmatrix}$$ , find the matrix $$(A+B)^T$$
• Part 3
Let matrix $$A = \begin{bmatrix} 0 & \dfrac{5}{3 \sqrt 5} & \dfrac{2}{3} \\ \dfrac{1}{\sqrt 5} & - \dfrac{4}{3 \sqrt 5} & \dfrac{2}{3} \\ \dfrac{2}{\sqrt 5} & \dfrac{2}{3 \sqrt 5} & - \dfrac{1}{3} \end{bmatrix}$$. Show that $$A A^T = A^T A = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ and therefore $$A^{-1} = A^T$$.