The elements of the product of two 3×3 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.
$$A= \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$$
$$B=\begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{pmatrix}$$
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.
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:
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 3×3 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.
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:
$$M=\begin{bmatrix} 15 & 11 & 5 \\ 22 & 12 & 9 \\22 & 18 & 7 \end{bmatrix}$$
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.
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:
$$M=\begin{bmatrix} 3 & 28 & 25 \\ 15 & 17 & 2 \\ -5 & -8 & -3 \end{bmatrix}$$
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:
$$M=\begin{bmatrix} -13 & 3 & 0 \\ -13 & 7 & -7 \\ -6 & -2 & 11 \end{bmatrix}$$
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:
$$M=\begin{bmatrix} 3& -3 & -21 \\ -3 & 6 & -13 \\ 0 & 6 & -20 \end{bmatrix}$$
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:
$$M=\begin{bmatrix} 16 & -18 & 24 \\ -10 & 23 & -54\\ -22 & 11 & -27 \end{bmatrix}$$
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:
$$M=\begin{bmatrix} 26& 21& 9\\-33& -23& -12\\22& -12& -13 \end{bmatrix}$$
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:
$$M=\begin{bmatrix} 10 & -14 & 30\\21& -26& 29\\57& -63& 3 \end{bmatrix}$$
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:
$$M=\begin{bmatrix} -37& 20& -3\\23& -13& -7\\23& -11& 23 \end{bmatrix} $$
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:
$$M=\begin{bmatrix} 24& -38& 59\\34& -38& 31\\-15& 10& -1 \end{bmatrix} $$
