# 2.5: The Transpose

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Another important operation on matrices is that of taking the transpose. For a matrix $$A$$, we denote the transpose of $$A$$ by $$A^T$$. Before formally defining the transpose, we explore this operation on the following matrix.

$\left[ \begin{array}{cc} 1 & 4 \\ 3 & 1 \\ 2 & 6 \end{array} \right] ^{T}= \ \ \left[ \begin{array}{ccc} 1 & 3 & 2 \\ 4 & 1 & 6 \end{array} \right] \nonumber$

What happened? The first column became the first row and the second column became the second row. Thus the $$3\times 2$$ matrix became a $$2\times 3$$ matrix. The number $$4$$ was in the first row and the second column and it ended up in the second row and first column.

The definition of the transpose is as follows.

## Definition $$\PageIndex{1}$$: The Transpose of a Matrix

Let $$A$$ be an $$m\times n$$ matrix. Then $$A^{T}$$, the transpose of $$A$$, denotes the $$n\times m$$ matrix given by

$A^{T} = \left[ a _{ij}\right] ^{T}= \left[ a_{ji} \right]\nonumber$

The $$\left( i, j \right)$$-entry of $$A$$ becomes the $$\left( j,i \right)$$-entry of $$A^T$$.

Consider the following example.

## Example $$\PageIndex{1}$$: The Transpose of a Matrix

Calculate $$A^T$$ for the following matrix

$A = \left[ \begin{array}{rrr} 1 & 2 & -6 \\ 3 & 5 & 4 \end{array} \right] \nonumber$

###### Solution

By Definition $$\PageIndex{1}$$, we know that for $$A = \left[ a_{ij} \right]$$, $$A^T = \left[ a_{ji} \right]$$. In other words, we switch the row and column location of each entry. The $$\left( 1, 2 \right)$$-entry becomes the $$\left( 2,1 \right)$$-entry.

Thus, $A^T = \left[ \begin{array}{rr} 1 & 3 \\ 2 & 5 \\ -6 & 4 \end{array} \right] \nonumber$

Notice that $$A$$ is a $$2 \times 3$$ matrix, while $$A^T$$ is a $$3 \times 2$$ matrix.

The transpose of a matrix has the following important properties.

## Lemma $$\PageIndex{1}$$: Properties of the Transpose of a Matrix

Let $$A$$ be an $$m\times n$$ matrix, $$B$$ an $$n\times p$$ matrix, and $$r$$ and $$s$$ scalars. Then

1. $\left(A^{T}\right)^{T} = A\nonumber$
2. $\left( AB\right) ^{T}=B^{T}A^{T} \nonumber$
3. $\left( rA+ sB\right) ^{T}=rA^{T}+ sB^{T} \nonumber$
Proof

First we prove 2. From Definition $$\PageIndex{1}$$,

\begin{aligned} \left(AB\right)^{T} &= \left[ (AB) _{ij} \right] ^{T}=\left[ (AB)_{ji} \right]=\sum_{k}a_{jk}b_{ki}= \sum_{k}b_{ki}a_{jk} \\[4pt] &= \sum_{k}\left[ b_{ik}\right]^{T}\left[ a_{kj}\right]^{T}=\left[ b_{ij}\right] ^{T} \left[ a_{ij}\right]^{T} = B^{T}A^{T} \end{aligned}

The proof of Formula 3 is left as an exercise.

The transpose of a matrix is related to other important topics. Consider the following definition.

## Definition $$\PageIndex{2}$$: Symmetric and Skew Symmetric Matrices

An $$n\times n$$ matrix $$A$$ is said to be symmetric if $$A=A^{T}.$$ It is said to be skew symmetric if $$A=-A^{T}.$$

We will explore these definitions in the following examples.

## Example $$\PageIndex{2}$$: Symmetric Matrices

Let

$A=\left[ \begin{array}{rrr} 2 & 1 & 3 \\ 1 & 5 & -3 \\ 3 & -3 & 7 \end{array} \right] \nonumber$

Use Definition $$\PageIndex{2}$$ to show that $$A$$ is symmetric.

###### Solution

By Definition $$\PageIndex{2}$$, we need to show that $$A = A^T$$. Now, using Definition $$\PageIndex{1}$$,

$A^{T} = \left[ \begin{array}{rrr} 2 & 1 & 3 \\ 1 & 5 & -3 \\ 3 & -3 & 7 \end{array} \right]\nonumber$

Hence, $$A = A^{T}$$, so $$A$$ is symmetric.

## Example $$\PageIndex{3}$$: A Skew Symmetric Matrix

Let

$A=\left[ \begin{array}{rrr} 0 & 1 & 3 \\ -1 & 0 & 2 \\ -3 & -2 & 0 \end{array} \right] \nonumber$

Show that $$A$$ is skew symmetric.

###### Solution

$A^{T} = \left[ \begin{array}{rrr} 0 & -1 & -3\\ 1 & 0 & -2\\ 3 & 2 & 0 \end{array} \right] \nonumber$

You can see that each entry of $$A^T$$ is equal to $$-1$$ times the same entry of $$A$$. Hence, $$A^{T} = - A$$ and so by Definition $$\PageIndex{2}$$, $$A$$ is skew symmetric.

This page titled 2.5: The Transpose is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Ken Kuttler (Lyryx) via source content that was edited to the style and standards of the LibreTexts platform.