## Illustrating matrix transpose rules in matrix multiplication

August 01, 2011 at 01:34 PM | categories: linear algebra | View Comments

# Illustrating matrix transpose rules in matrix multiplication

John Kitchin

## Contents

## Rules for transposition

Here are the four rules for matrix multiplication and transposition

1.

2.

3.

4.

reference: Chapter 7.2 in Advanced Engineering Mathematics, 9th edition. by E. Kreyszig.

## The transpose in Matlab

there are two ways to get the transpose of a matrix: with a notation, and with a function

A = [[5 -8 1]; [4 0 0]]

A = 5 -8 1 4 0 0

function

transpose(A)

ans = 5 4 -8 0 1 0

notation

A.' % note, these functions only provide the non-conjugate transpose. If your % matrices are complex, then you want the ctranspose function, or the % notation A' (no dot before the apostrophe). For real matrices there is no % difference between them. % below we illustrate each rule using the different ways to get the % transpose.

ans = 5 4 -8 0 1 0

## Rule 1

```
m1 = (A.').'
A
all(all(m1 == A)) % if this equals 1, then the two matrices are equal
```

m1 = 5 -8 1 4 0 0 A = 5 -8 1 4 0 0 ans = 1

## Rule 2

```
B = [[3 4 5];
[1 2 3]];
m1 = transpose(A+B)
m2 = transpose(A) + transpose(B)
all(all(m1 == m2)) % if this equals 1, then the two matrices are equal
```

m1 = 8 5 -4 2 6 3 m2 = 8 5 -4 2 6 3 ans = 1

## Rule 3

```
c = 2.1;
m1 = transpose(c*A)
m2 = c*transpose(A)
all(all(m1 == m2)) % if this equals 1, then the two matrices are equal
```

m1 = 10.5000 8.4000 -16.8000 0 2.1000 0 m2 = 10.5000 8.4000 -16.8000 0 2.1000 0 ans = 1

## Rule 4

```
B = [[0 2];
[1 2];
[6 7]]
m1 = (A*B).'
m2 = B.'*A.'
all(all(m1 == m2)) % if this equals 1, then the two matrices are equal
```

B = 0 2 1 2 6 7 m1 = -2 0 1 8 m2 = -2 0 1 8 ans = 1

m3 = A.'*B.' % you can see m3 has a different shape than m1, so there is no way they can % be equal.

m3 = 8 13 58 0 -8 -48 0 1 6

% categories: Linear algebra % tags: math % post_id = 552; %delete this line to force new post;