Lesson Objectives
• Learn how to find the transpose of a matrix

## How to Find the Transpose of a Matrix

In this lesson, we will learn how to find the transpose of a matrix. This is done by swapping or interchanging the rows and columns. Let's look at an example.
Example #1: Find the transpose. $$A=\left[ \begin{array}{ccc}3&1&6\\ 5&2&1\\-5&-7&1\end{array}\right]$$ We will use AT to stand for the transpose of matrix A. To find the transpose of matrix A, we just take row 1 and make it column 1, then take row 2 and make it column 2, and then finally take row 3 and make it column 3. $$A^T=\left[ \begin{array}{ccc}3&5&-5\\ 1&2&-7\\6&1&1\end{array}\right]$$

#### Skills Check:

Example #1

Find the transpose of A. $$A=\left[ \begin{array}{cc}2&0\\ -1&3\end{array}\right]$$

A
$$A^T=\left[ \begin{array}{cc}1&0\\ 3&3\end{array}\right]$$
B
$$A^T=\left[ \begin{array}{cc}2&-1\\ 0&3\end{array}\right]$$
C
$$A^T=\left[ \begin{array}{cc}-1&2\\ 3&0\end{array}\right]$$
D
$$A^T=\left[ \begin{array}{cc}2&-1\\ 3&-1\end{array}\right]$$
E
$$A^T=\left[ \begin{array}{cc}-1&3\\ 2&1\end{array}\right]$$

Example #2

Find the transpose of A. $$A=\left[ \begin{array}{cc}1&3\\ 5&9\end{array}\right]$$

A
$$A^T=\left[ \begin{array}{cc}5&1\\ 3&9\end{array}\right]$$
B
$$A^T=\left[ \begin{array}{cc}3&1\\ 5&9\end{array}\right]$$
C
$$A^T=\left[ \begin{array}{cc}9&3\\ 1&5\end{array}\right]$$
D
$$A^T=\left[ \begin{array}{cc}1&5\\ 3&9\end{array}\right]$$
E
$$A^T=\left[ \begin{array}{cc}-1&-3\\ -5&-9\end{array}\right]$$