Multiplication of 3x3 matrices is an operation that has many applications in physics, engineering, and other fields. Each element of the resulting matrix is found by multiplying each row of the first matrix by the corresponding columns of the second matrix and adding the products.
In this article, we will learn how to solve 3x3 matrix multiplication. We will start with a step-by-step process to multiply two 3x3 matrices. Then, we will solve several exercises to apply this process.
How to multiply 3x3 matrices?
The elements of the product of two 3x3 matrices are found by multiplying the elements of each row of the first matrix by the corresponding elements of each column of the second matrix.
The following steps are a guide on how to multiply 3x3 matrices:
Step 1: Write the matrices in the following way:
$$A= \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$$
$$B=\begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{pmatrix}$$
Step 2: Starting with the first row of the first matrix, multiply each element by the corresponding element in the first column of the second matrix and add the products.
This way we will get the first element of the product matrix:
$$c_{11} = a_{11} \times b_{11} + a_{12} \times b_{21} + a_{13} \times b_{31}$$
Step 3: Repeat step 2 for the remaining elements of the first row of the first matrix and the remaining columns of the second matrix. This way, we will obtain the first row of the product matrix:
$$c_{11} = a_{11} \times b_{11} + a_{12} \times b_{21} + a_{13} \times b_{31}$$
$$c_{12} =a_{11} \times b_{12} + a_{12} \times b_{22} + a_{13} \times b_{32}$$
$$ c_{13}=a_{11} \times b_{13} + a_{12} \times b_{23} + a_{13} \times b_{33}$$
Step 4: Repeat steps 2 and 3 for the remaining rows of the first matrix, using each row to produce a corresponding row in the product matrix.
The resulting matrix is the product of the two 3x3 matrices.
$$\begin{pmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{pmatrix}$$
Note: Matrix multiplication is not commutative, which means that the order of the matrices matters. That is, AxB is not necessarily the same as BxA.
Solved exercises on multiplication of 3x3 matrices
EXAMPLE 1
Find the product M resulting from the multiplication of matrices A and B:
$$A=\begin{bmatrix} 2 & 1 & 3 \\ 3 & 4 & 1 \\ 5 & 2 & 3 \end{bmatrix}$$
$$B=\begin{bmatrix} 1 & 2 & 0 \\ 4 & 1 & 2 \\ 3 & 2 & 1 \end{bmatrix}$$
Solution
The matrix M is the result of the multiplication A x B.
The term $latex m_{11}$ is found by multiplying the elements of the first row of the matrix A by the elements of the first column of the matrix B. Then, we have:
$latex m_{11}=2 \times 1 + 1 \times 4+3 \times 3$
$latex m_{11}=2 + 4+9=15$
For the term $latex m_{12}$, we multiply the elements of the first row of the matrix A by the elements of the second column of the matrix B:
$latex m_{12}=2 \times 2 + 1 \times 1+3 \times 2$
$latex m_{12}=4 + 1+6=11$
The term $latex m_{13}$ is found by taking the first row of matrix A and the third column of matrix B:
$latex m_{13}=2 \times 0 + 1 \times 2+3 \times 1$
$latex m_{13}= 0 + 2+3=5$
The term $latex m_{21}$ is found by taking the second row of matrix A and the first column of matrix B:
$latex m_{21}=3 \times 1 + 4 \times 4+1 \times 3$
$latex m_{21}=3 + 16+ 3=22$
For the $latex m_{22}$ term, we multiply the elements of the second row of matrix A by the corresponding elements of the second column of matrix B:
$latex m_{22}=3 \times 2 + 4 \times 1+1 \times 2$
$latex m_{22}=6 + 4+ 2=12$
For the term $latex m_{23}$, we take the second row of matrix A and the third column of matrix B:
$latex m_{23}=3 \times 0 + 4 \times 2+1 \times 1$
$latex m_{21}=0 + 8+ 1=9$
For the term $latex m_{31}$, we take the third row of matrix A and the first column of matrix B:
$latex m_{31}=5 \times 1 + 2 \times 4+3 \times 3$
$latex m_{31}=5 + 8+ 9=22$
The term $latex m_{32}$ is found with the third row of matrix A and the second column of matrix B:
$latex m_{32}=5 \times 2 + 2 \times 1+3 \times 2$
$latex m_{32}=10 + 2+ 6=18$
The term $latex m_{33}$ is found with the third row of matrix A and the third column of matrix B:
$latex m_{33}=5 \times 0 + 2 \times 2+3 \times 1$
$latex m_{33}=0 + 4+ 3=7$
So, we have:
$$M=\begin{bmatrix} 15 & 11 & 5 \\ 22 & 12 & 9 \\22 & 18 & 7 \end{bmatrix}$$
EXAMPLE 2
Find the product of the multiplication B×A.
$$A=\begin{bmatrix} 2 & 1 & 3 \\ 3 & 4 & 1 \\ 5 & 2 & 3 \end{bmatrix}$$
$$B=\begin{bmatrix} 1 & 2 & 0 \\ 4 & 1 & 2 \\ 3 & 2 & 1 \end{bmatrix}$$
Solution
The multiplication of the matrices can be written in the following form to simplify the process:
$$M=\begin{bmatrix} (1\times 2)+(2\times 3)+(0\times 5) & (1\times 1)+(2\times 4)+(0\times 2) & (1\times 3)+(2\times 1)+(0\times 3) \\ (4\times 2)+(1\times 3)+(2\times 5) & (4\times 1)+(1\times 4)+(2\times 2) & (4\times 3)+(1\times 1)+(2\times 3) \\ (3\times 2)+(2\times 3)+(1\times 5) & (3\times 1)+(2\times 4)+(1\times 2) & (3\times 3)+(2\times 1)+(1\times 3) \end{bmatrix}$$
$$M=\begin{bmatrix}8 & 9 & 5 \\21& 12 & 19 \\17 & 13 & 14\end{bmatrix}$$
Comparing examples 1 and 2, we see that multiplication A×B is not equal to multiplication B×A. Order is important when multiplying matrices.
EXAMPLE 3
Solve the multiplication A×B using the following matrices:
$$A=\begin{bmatrix} -2 & 3 & 4 \\ 2 & -3 & 5 \\ 0 & 3 & -4 \end{bmatrix}$$
$$B=\begin{bmatrix} 4 & 2 & -2 \\ 1 & 4 & 3 \\ 2 & 5 & 3 \end{bmatrix}$$
Solution
Find each element of the resulting matrix by performing the required operations:
$$M=\begin{bmatrix} (-2\times 4)+(3\times 1)+(4\times 2) & (-2\times 2)+(3\times 4)+(4\times 5) & (-2\times -2)+(3\times 3)+(4\times 3) \\ (2\times 4)+(-3\times 1)+(5\times 2) & (2\times 2)+(-3\times 4)+(5\times 5) & (2\times -2)+(-3\times 3)+(5\times 3) \\ (0\times 4)+(3\times 1)+(-4\times 2) & (0\times 2)+(3\times 4)+(-4\times 5) & (0\times -2)+(3\times 3)+(-4\times 3) \end{bmatrix}$$
Simplifying this, we have:
$$M=\begin{bmatrix} 3 & 28 & 25 \\ 15 & 17 & 2 \\ -5 & -8 & -3 \end{bmatrix}$$
EXAMPLE 4
Find the matrix M that is equal to A×B.
$$A=\begin{bmatrix} 3 & -2 & -1 \\ 3 & -3 & 2 \\ 2 & 1 & -5 \end{bmatrix}$$
$$B=\begin{bmatrix} -2 & 3 & 0 \\ 3 & 2 & 1 \\ 1 & 2 & -2 \end{bmatrix}$$
Solution
We can find each element of matrix M by multiplying the elements of each row of matrix A by the corresponding elements of each column of matrix B:
$$M=\begin{bmatrix} (3\times -2)+(-2\times 3)+(-1\times 1) & (3\times 3)+(-2\times 2)+(-1\times 2) & (3\times 0)+(-2\times 1)+(-1\times -2) \\ (3\times -2)+(-3\times 3)+(2\times 1) & (3\times 3)+(-3\times 2)+(2\times 2) & (3\times 1)+(-3\times 2)+(2\times -2) \\ (2\times -2)+(1\times 3)+(-5\times 1) & (2\times 2)+(1\times 2)+(-5\times 2) & (2\times 0)+(1\times 1)+(-5\times -2) \end{bmatrix}$$
Now, we simplify the operations and we have:
$$M=\begin{bmatrix} -13 & 3 & 0 \\ -13 & 7 & -7 \\ -6 & -2 & 11 \end{bmatrix}$$
EXAMPLE 5
Multiply the following matrices to find the product A×B:
$$A=\begin{bmatrix} 5 & -3 & -4 \\ 1 & -2 & -2 \\ 3 & -4 & -3 \end{bmatrix}$$
$$B=\begin{bmatrix} 3 & -4 & 1 \\ 0 & -3 & 2 \\ 3 & -2 & 5 \end{bmatrix}$$
Solution
Perform the following operations to find the elements of the resulting matrix:
$$M=\begin{bmatrix} (5\times 3)+(-3\times 0)+(-4\times 3) & (5\times -4)+(-3\times -3)+(-4\times -2) & (5\times 1)+(-3\times 2)+(-4\times 5) \\ (1\times 3)+(-2\times 0)+(-2\times 3) & (1\times -4)+(-2\times -3)+(-2\times -2) & (1\times 1)+(-2\times 2)+(-2\times 5) \\ (3\times 3)+(-4\times 0)+(-3\times 3) & (3\times -4)+(-4\times -3)+(-3\times -2) & (3\times 1)+(-4\times 2)+(-3\times 5) \end{bmatrix}$$
Simplifying, we have:
$$M=\begin{bmatrix} 3& -3 & -21 \\ -3 & 6 & -13 \\ 0 & 6 & -20 \end{bmatrix}$$
EXAMPLE 6
Find product A×B:
$$A=\begin{bmatrix} 2 & 6 & 0 \\ 5 & -7 & -1 \\ 2 & -1 & -4 \end{bmatrix}$$
$$B=\begin{bmatrix} 2 & 0 & -3 \\ 2 & -3 & 5 \\ 6 & -2 & 4 \end{bmatrix}$$
Solution
Find the elements of the resulting matrix as follows:
$$M=\begin{bmatrix} (2\times 2)+(6\times 2)+(0\times 6) & (2\times 0)+(6\times -3)+(0\times -2) & (2\times -3)+(6\times 5)+(0\times 4) \\ (5\times 2)+(-7\times 2)+(-1\times 6) & (5\times 0)+(-7\times -3)+(-1\times -2) & (5\times -3)+(-7\times 5)+(-1\times 4) \\ (2\times 2)+(-1\times 2)+(-4\times 6) & (2\times 0)+(-1\times -3)+(-4\times -2) & (2\times -3)+(-1\times 5)+(-4\times 4) \end{bmatrix} $$
When we solve this, we have:
$$M=\begin{bmatrix} 16 & -18 & 24 \\ -10 & 23 & -54\\ -22 & 11 & -27 \end{bmatrix}$$
EXAMPLE 7
Find the product of the multiplication A×B:
$$A=\begin{bmatrix} 2&5&-7\\-3&-5&6\\0&3&2 \end{bmatrix}$$
$$B=\begin{bmatrix} 5&5&5\\6&-2&-3\\2&-3&-2 \end{bmatrix}$$
Solution
The elements of the resulting matrix are found as follows:
$$M=\begin{bmatrix} (2\times 5)+(5\times 6)+(-7\times 2) & (2\times 5)+(5\times -2)+(-7\times -3) & (2\times 5)+(5\times -3)+(-7\times -2) \\ (-3\times 5)+(-5\times 6)+(6\times 2) & (-3\times 5)+(-5\times -2)+(6\times -3) & (-3\times 5)+(-5\times -3)+(6\times -2) \\ (0\times 5)+(3\times 6)+(2\times2) & (0\times 5)+(3\times -2)+(2\times -3) & (0\times 5)+(3\times -3)+(2\times -2) \end{bmatrix}$$
And the matrix product is:
$$M=\begin{bmatrix} 26& 21& 9\\-33& -23& -12\\22& -12& -13 \end{bmatrix}$$
EXAMPLE 8
Solve the multiplication A×B.
$$A=\begin{bmatrix} 4&2&-3\\5&-1&1\\6&7&3 \end{bmatrix}$$
$$B=\begin{bmatrix} 4&-5&6\\3&-3&-3\\4&-4&-4 \end{bmatrix}$$
Solution
To find each element of the resulting matrix, we have:
$$M=\begin{bmatrix} (4\times 4)+(2\times 3)+(-3\times 4) & (4\times -5)+(2\times -3)+(-3\times -4) & (4\times 6)+(2\times -3)+(-3\times -4) \\ (5\times 4)+(-1\times 3)+(1\times 4) & (5\times -5)+(-1\times -3)+(1\times -4) & (5\times 6)+(-1\times -3)+(1\times -4) \\ (6\times 4)+(7\times3)+(3\times 4) & (6\times -5)+(7\times -3)+(3\times -4) & (6\times 6)+(7\times -3)+(3\times -4) \end{bmatrix}$$
And simplifying:
$$M=\begin{bmatrix} 10 & -14 & 30\\21& -26& 29\\57& -63& 3 \end{bmatrix}$$
EXAMPLE 9
Find the product of multiplying A by B:
$$A=\begin{bmatrix} -5&4&-3\\3&-5&3\\4&2&-1 \end{bmatrix}$$
$$B=\begin{bmatrix} 6&-3&4\\2&-1&2\\5&-3&-3 \end{bmatrix}$$
Solution
We have the following:
$$M=\begin{bmatrix} (-5\times 6)+(4\times 2)+(-3\times 5) & (-5\times -3)+(4\times -1)+(-3\times -3) & (-5\times 4)+(4\times 2)+(-3\times -3) \\ (3\times 6)+(-5\times 2)+(3\times 5) & (3\times -3)+(-5\times -1)+(3\times -3) & (3\times 4)+(-5\times 2)+(3\times -3) \\ (4\times 6)+(2\times2)+(-1\times 5) & (4\times -3)+(2\times -1)+(-1\times -3) & (4\times 4)+(2\times -3)+(-1\times -3) \end{bmatrix}$$
Simplifying, we have:
$$M=\begin{bmatrix} -37& 20& -3\\23& -13& -7\\23& -11& 23 \end{bmatrix} $$
EXAMPLE 10
Find the product A x B of the following matrices:
$$A=\begin{bmatrix} 6&3&-5\\2&5&3\\-3&4&-4 \end{bmatrix}$$
$$B=\begin{bmatrix} 5&-6&7\\3&-4&4\\3&-2&-1 \end{bmatrix}$$
Solution
We have the following:
$$M=\begin{bmatrix} (6\times 5)+(3\times 3)+(-5\times 3) & (6\times -6)+(3\times -4)+(-5\times -2) & (6\times 7)+(3\times 4)+(-5\times -1) \\ (2\times 5)+(5\times 3)+(3\times 3) & (2\times -6)+(5\times -4)+(3\times -2) & (2\times 7)+(5\times 4)+(3\times -1) \\ (-3\times 5)+(4\times 3)+(-4\times 3) & (-3\times -6)+(4\times -4)+(-4\times -2) & (-3\times 7)+(4\times 4)+(-4\times -1) \end{bmatrix}$$
So, we have:
$$M=\begin{bmatrix} 24& -38& 59\\34& -38& 31\\-15& 10& -1 \end{bmatrix} $$
Multiplication of 3x3 matrices - Practice problems
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See also
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