How to Multiply Matrix? (+FREE Worksheet!)

Matrix Multiplication – Example 1:

\(\begin{bmatrix}-5 & -5 \\-1 & 2 \end{bmatrix}\)\(\begin{bmatrix}-2 & -3 \\3 & 5 \end{bmatrix}\)

Solution:

Multiply the rows of the first matrix by the columns of the second matrix. \(\begin{bmatrix}(-5)(-2)+(-5)(3) & (-5)(-3)+(-5)(5) \\(-1)(-2)+(2)(3) & (-1)(-3)+(2)(5) \end{bmatrix}= \begin{bmatrix}(10)+(-15) & (15)+(-25) \\(2)+(6) & (3)+(10) \end{bmatrix}=\begin{bmatrix}-5 & -10 \\8 & 13 \end{bmatrix}\)

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Matrix Multiplication – Example 2:

\(\begin{bmatrix}-4 & -6&-6 \\0 & 6&3 \end{bmatrix}\begin{bmatrix}0 \\-3 \\0 \end{bmatrix}\)

Solution:

Multiply the rows of the first matrix by the columns of the second matrix. \(\begin{bmatrix}(-4)(0)+(-6)(-3)+(-6)(0) \\(0)(0)+(6)(-3)+(3)(0) \end{bmatrix}=\begin{bmatrix}0+18+0 \\0-18+0 \end{bmatrix}=\begin{bmatrix}18 \\-18 \end{bmatrix}\)

Matrix Multiplication – Example 3:

\(\begin{bmatrix}1 & 3 \\2 & 4 \end{bmatrix}\)\(\begin{bmatrix}2 &4 \\-2 & 1 \end{bmatrix}\)

Solution:

\(\begin{bmatrix}(1) (2)+(3)(-2) & (1) (4)+(3) (1) \\(2) (2)+ (4)(-2) & (2) (4)+(4) (1) \end{bmatrix}=\begin{bmatrix}(2)+(-6) & (4)+(3) \\(4)+ (-8) & (8)+(4) \end{bmatrix}=\begin{bmatrix}-4 & 7 \\-4 & 12 \end{bmatrix}\)

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Matrix Multiplication – Example 4:

\(\begin{bmatrix}2 & -1&-1 \\3 & 1&5 \end{bmatrix}\begin{bmatrix}-2 \\-1 \\4 \end{bmatrix}\)

Solution:

Multiply the rows of the first matrix by the columns of the second matrix. \(\begin{bmatrix}(2)(-2)+(-1)(-1)+(-1) (4)\\(3)(-2)+(1)(-1)+(5) (4) \end{bmatrix}=\begin{bmatrix}(-4)+(1)+(-4)\\(-6)+(-1)+(20) \end{bmatrix}=\begin{bmatrix}-7 \\13 \end{bmatrix}\)

Exercises for Multiplying Matrix

Matrix Multiplication

Solve.

1. \(\color{blue}{\begin{bmatrix}0 & 2 \\-2 & -5 \end{bmatrix}\begin{bmatrix}6 & -6 \\3 & 0 \end{bmatrix}}\)
2. \(\color{blue}{\begin{bmatrix}3 & -1 \\-3 & 6\\-6&-6 \end{bmatrix}\begin{bmatrix}-1 & 6 \\5 & 4\end{bmatrix}}\)
3. \(\color{blue}{\begin{bmatrix}0 & 5 \\-3 & 1\\-5&1 \end{bmatrix}\begin{bmatrix}-4 & 4 \\-2 & -4\end{bmatrix}}\)
4. \(\color{blue}{\begin{bmatrix}5 & 3&5 \\1 & 5&0 \end{bmatrix}\begin{bmatrix}-4 & 2 \\-3 & 4\\3&-5 \end{bmatrix}}\)
5. \(\color{blue}{\begin{bmatrix}4 & 5 \\-4 & 6\\-5&-6 \end{bmatrix}\begin{bmatrix}4 & 6 \\6& 2\\-4&1 \end{bmatrix}}\)
6. \(\color{blue}{\begin{bmatrix}-2 & -6 \\-4 & 3\\5&0 \\4&-6\end{bmatrix}\begin{bmatrix}2 & -2&2 \\-2 &0&-3 \end{bmatrix}}\)

1. \(\color{blue}{\begin{bmatrix}6 & 0 \\-27 & 12 \end{bmatrix}}\)
2. \(\color{blue}{\begin{bmatrix}-8 & 14 \\33 & 6\\ -24&-60\end{bmatrix}}\)
3. \(\color{blue}{\begin{bmatrix}-10 & -20 \\10 & -16\\ 18&-24\end{bmatrix}}\)
4. \(\color{blue}{\begin{bmatrix}-14 & -3 \\-19 & 22 \end{bmatrix}}\)
5. \(\color{blue}{Undefined}\)
6. \(\color{blue}{\begin{bmatrix}8 & 4&14\\-14 & 8&-17\\10&-10&10 \\20&-8&26\end{bmatrix}}\)

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