Multiplication of 3x3 matrix

This article will be about the multiplication of matrices of order 3x3, examples and the procedure to get the product.

How to multiply 3x3 matrices

In this article we are going to develop various examples of how to multiply a 3x3 matrix. When we multiply 2 matrices it is important to check that one of the matrices have the same amount of rows as the columns of the other matrix, this means that if one of the matrices have 3 rows, the other matrix must have 3 columns, otherwise, we cannot multiply the matrices.

Based on the previous explanation, we can always multiply two 3x3 matrices, because the previous rule is always accomplished. The result of a multiplication between two 3x3 matrices is going to be another matrix of the same order.

A11 A12 A13
A21 A22 A23
A31 A32 A33
B11 B12 B13
B21 B22 B23
B31 B32 B33

The multiplication between matrices is done by multiplying each row of the first matrix with every column of the second matrix, and then adding the results, just like in the next example.

  • Row 1
  • C11 = (A11 * B11) + (A12 * B21) + (A13 * B31)
  • C12 = (A11 * B12) + (A12 * B22) + (A13 * B32)
  • C12 = (A11 * B13) + (A12 * B23) + (A13 * B33)
  • Row 2
  • C21 = (A21 * B11) + (A22 * B21) + (A23 * B31)
  • C22 = (A21 * B12) + (A22 * B22) + (A23 * B32)
  • C22 = (A21 * B13) + (A22 * B23) + (A23 * B33)
  • Row 3
  • C31 = (A31 * B11) + (A32 * B21) + (A33 * B31)
  • C32 = (A31 * B12) + (A32 * B22) + (A33 * B32)
  • C32 = (A31 * B13) + (A32 * B23) + (A33 * B33)

Now seeing this, we are going to make an example of how to multiply two 3x3 matrices

Matrix A

2 3 1
7 4 1
9 -2 1

Matrix B

9 -2 -1
5 7 3
8 1 0
  • Now we find the resultant matrix
  • Row 1
  • C11= (2*9) + (3*5) + (1*8)
  • C11= 18 + 15 + 8
  • C11= 41
  • C12= (2*-2) + (3*7) + (1*1)
  • C12= -4 + 21 + 1
  • C12= 18
  • C13= (2*-1) + (3*3) + (1*0)
  • C13= -2 + 9 + 0
  • C13= 7
  • Row 2
  • C21= (7*9) + (4*5) + (1*8)
  • C21= 63 + 20 + 8
  • C21= 91
  • C22= (7*-2) + (4*7) + (1*1)
  • C22= -14 + 28 + 1
  • C22= 15
  • C23= (7*-1) + (4*3) + (1*0)
  • C23= -7 + 12 + 0
  • C23= 5
  • Row 3
  • C31= (9*9) + (-2*5) + (1*8)
  • C31= 81 -10 + 8
  • C31= 79
  • C32= (9*-2) + (-2*7) + (1*1)
  • C32= -18 -14 + 1
  • C32= -31
  • C33= (9*-1) + (-2*3) + (1*0)
  • C33= -9 -6 + 0
  • C33= -15

Resultant matrix

41 18 7
91 15 5
79 -31 -15

Examples of multiplication of 3x3 matrices

Example 1: Multiply the following 3x3 matrices.

Matrix A

5 5 0
2 2 1
3 3 2

Matrix B

0 -1 -1
-1 0 -1
0 0 -1

Now with the matrix we are going to multiply, we are going to multiply every row of the first matrix with every column of the second matrix, and this way we are going to find every position in the resultant matrix (C)

  • First we solve the first row
  • C11 = (5*0) + (5*-1) + (0*0)
  • C11 = 0 - 5 + 0
  • C11 = -5
  • C12 = (5*-1) + (5*0) + (0*0)
  • C12 = -2 + 0 + 0
  • C12 = -2
  • C13 = (5*-1) + (5*-1) + (0*-1)
  • C13 = -2 - 5 + 0
  • C13 = -7
  • Now the second row
  • C21 = (2*0)+(2*-1)+(1*0)
  • C21 = 0 -2 0
  • C21 = -2
  • C22 = (2*-1)+(2*0)+(1*0)
  • C22 = -2 + 0 + 0
  • C22 = -2
  • C23 = (2*-1)+(2*-1)+(1*-1)
  • C23 = -2 - 2 -1
  • C23 = -5
  • And now the third row
  • C31 = (3*0)+(3*-1)+(2*0)
  • C31 = 0 -3 + 0
  • C31 = -3
  • C32 = (3*-1)+(3*0)+(2*0)
  • C32 = -3 + 0 + 0
  • C32 = -3
  • C33 = (3*-1)+(3*-1)+(2*-1)
  • C33 = -3 - 3 -2
  • C33 = -8

Resultant matrix (C)

-5 -2 -7
-2 -2 -5
-3 -3 -8

Example 2: ¿What is the resultant matrix when we multiply the next two matrices?

Matrix A

2 1 5
2 10 5
3 1 4

Matrix B

8 7 1
4 2 7
2 3 5

Now we multiply every row and column

  • Row 1
  • C11 = (2*8) + (1*4) + (5*2)
  • C11 = 16 + 4 + 10
  • C11 = 30
  • C12 = (2*7) + (1*2) + (5*3)
  • C12 = 14 + 2 + 15
  • C12 = 31
  • C13 = (2*1) + (1*7) + (5*5)
  • C13 = 2 + 7 + 25
  • C13 = 34
  • Row 2
  • C21 = (2*8)+(10*4)+(5*2)
  • C21 = 16 + 40 + 10
  • C21 = 66
  • C22 = (2*7) + (10*2) + (5*3)
  • C22 = 14 + 20 + 15
  • C22 = 49
  • C23 = (2*1) + (10*7) + (5*5)
  • C23 = 2 + 70 + 25
  • C23 = 97
  • Row 3
  • C31 = (3*8) + (1*4) + (4*2)
  • C31 = 24 + 4 + 8
  • C31 = 36
  • C32 = (3*7) + (1*2) + (4*3)
  • C32 = 21 + 2 + 12
  • C32 = 35
  • C33 = (3*1) + (1*7) + (4*5)
  • C33 = 3 + 7 + 20
  • C33 = 30

Resultant matrix (C)

30 31 34
66 49 97
36 35 30

Example 3: Multiply A x B and find the result "C"

Matrix A

3 1 1
8 9 4
2 5 6

Matrix B

3 5 7
3 1 8
10 5 2
  • Solve row 1
  • C11 = (3*3) + (1*3) + (1*10)
  • C11 = 9 + 3 + 10
  • C11 = 22
  • C12 = (3*5) + (1*1) + (1*5)
  • C12 = 15 + 1 + 5
  • C12 = 21
  • C13 = (3*7) + (1*8) + (1*2)
  • C13 = 21 + 8 + 2
  • C13 = 31
  • Solve row 2
  • C21 = (8*3) + (9*3) + (4*10)
  • C21 = 24 + 27 + 40
  • C21 = 91
  • C22 = (8*5) + (9*1) + (4*5)
  • C22 = 40 + 9 + 20
  • C22 = 69
  • C23 = (8*7) + (9*8) + (4*2)
  • C23 = 56 + 72 + 8
  • C23 = 136
  • Solve row 3
  • C31 = (2*3) + (5*3) + (6*10)
  • C31 = 6 + 15 + 60
  • C31 = 81
  • C32 = (2*5) + (5*1) + (6*5)
  • C32 = 10 + 5 + 30
  • C32 = 45
  • C33 = (2*7) + (5*8) + (6*2)
  • C33 = 14 + 40 + 12
  • C33 = 66

Resultant matrix (C)

22 21 31
91 69 136
81 45 66