Matrix Multiplication

Here is a step by step guide to multiply matrices. The exercises can help you measure your knowledge of matrix multiplication.

Step by step guide to multiply matrices

• Step 1: Make sure that it’s possible to multiply the two matrices (the number of columns in the 1st one should be the same as the number of rows in the second one.)
• Step 2: The elements of each row of the first matrix should be multiplied by the elements of each column in the second matrix.
• Step 3: Add the products.

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. \(\begin{bmatrix}(-5)(-2)+(-5).3 & (-5)(-3)+(-5).5 \\(-1)(-2)+2.3 & (-1)(-3)+2.5 \end{bmatrix}=\begin{bmatrix}-5 & -10 \\8 & 13 \end{bmatrix}\)

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. \(\begin{bmatrix}(-4).0+(-6)(-3)+(-6).0 \\0.0+6(-3)+3.0 \end{bmatrix}=\begin{bmatrix}18 \\-18 \end{bmatrix}\)

Example 3:

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

Solution:

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

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. \(\begin{bmatrix}2(-2)+(-1)(-1)+(-1) .4\\3(-2)+1 .(-1)+5 .4 \end{bmatrix}=\begin{bmatrix}-7 \\13 \end{bmatrix}\)

Exercises

Solve.

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

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