12.1: Basic Matrix Multiplication

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Carey Smith
Oxnard College

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By Carey Smith

Matrix Multiplication Summary:

A matrix times a vector (3x3 case):

\( A*B =
\begin{matrix}
|a & b & c| \\
|p & q & r| \\
|u & v & w|
\end{matrix} * \begin{matrix}
|x| \\
|y| \\
|z|
\end{matrix} = \begin{matrix}
|a*x & b*y & c*z| \\
|p*x & q*y & r*z| \\
|u*x & v*y & w*z|
\end{matrix}
\)

Matrices do not need to be square, but the number of columns in A needs to equal the number of rows in B.

Matrix multiplication is generally not commutative. That is A*B ≠ B*A, in general.

Watch this video about multiplying matrices:

Matlab Video Tutorial: Multiplying Matrices and Vectors, https://www.youtube.com/watch?v=sgzFn42jU6Y

Example \(\PageIndex{1}\) Matrix Multiply 3x3 times a vector

A = [3 2 1;

2 1 0

1 0 -1]

B = [4

A*B

Solution

A*B =

28
13
-2

% = [28 %(28 = 3*4 + 2*5 + 1*6)

% 13 %(13 = 2*4 + 1*5 + 0*6)

% -2] %(-2 = 1*4 + 0*5 + -1*6)

Exercise \(\PageIndex{1}\) Matrix Multiplication True/False

1. True or False: These 2 matrices (which have 3 rows and 2 columns) can be multiplied together:

[ 1 5 ] [ 9 5 ]

A = | 2 9 | B = | 8 9 |

[ 3 4 ] [ 7 5 ]

2. True or False: These 2 matrices can be multiplied together:

[ 1 5 ]

A = | 2 9 |; B = [ 1 3 2 ]

[ 3 4 ] [ 4 6 9 ]

Answer

1. No, because the number of columns in A (2) does not equal the number of rows in B (3).

1. Yes, because the number of columns in A (2) does equal the number of rows in B (2).

Exercise \(\PageIndex{2}\) Matrix Multiplication Calculation

1. What is A*B ?

A = [1 4 3 -5]

B = [ 2 12

3 1]

(Do a matrix multiply, not an element-by element multiply)

Answer

A*B = [14 16

-9 31]

A matrix with ones on the diagonal and zeros elsewhere is called the Identity matrix:

I = eye(3) = [1 0 0

0 1 0

0 0 1]

I*A = A

A*I = A

For more information, see: https://en.Wikipedia.org/wiki/Matrix_multiplication