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

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Sum 3x3 matrix Subtraction of matrices Sum 2x2 matrix Multiply matrix by scalar