Matrix Multiplication Examples- Practice Problems

Matrix Multiplication Examples That Actually Make Sense

Most textbooks turn a simple concept into a nightmare. Here's the truth: matrix multiplication is just systematic multiplication and addition. Once you see the pattern, you'll wonder why anyone made it seem complicated.

What You Need Before Multiplying

Two matrices can multiply only if the columns of the first matrix equal the rows of the second matrix.

Say you have a 2×3 matrix and a 3×4 matrix. The inner numbers match (both are 3), so multiplication works. The result will be a 2×4 matrix.

If those numbers don't match, stop. There's no point trying to multiply.

How to Multiply Two Matrices

Here's the process:

• Take each row of the first matrix
• Multiply it element-by-element with each column of the second matrix
• Add up those products
• That sum becomes one entry in your result matrix

That's it. Every single entry comes from one row times one column.

Example 1: Simple 2×2 Multiplication

Multiply:

A = [2, 3]
　　[1, 4]

B = [5, 1]
　　[2, 6]

Step 1: Find entry (1,1)

Row 1 of A = [2, 3]
Column 1 of B = [5, 2]

2×5 + 3×2 = 10 + 6 = 16

Step 2: Find entry (1,2)

Row 1 of A = [2, 3]
Column 2 of B = [1, 6]

2×1 + 3×6 = 2 + 18 = 20

Step 3: Find entry (2,1)

Row 2 of A = [1, 4]
Column 1 of B = [5, 2]

1×5 + 4×2 = 5 + 8 = 13

Step 4: Find entry (2,2)

Row 2 of A = [1, 4]
Column 2 of B = [1, 6]

1×1 + 4×6 = 1 + 24 = 25

Result:

[16, 20]
[13, 25]

Example 2: Different Dimensions

Multiply:

A = [1, 2, 3]
　　[4, 5, 6]　　This is a 2×3 matrix

B = [7, 8]
　　[9, 10]
　　[11, 12]　　This is a 3×2 matrix

The result will be 2×2.

Entry (1,1): Row 1 × Column 1
(1×7) + (2×9) + (3×11) = 7 + 18 + 33 = 58

Entry (1,2): Row 1 × Column 2
(1×8) + (2×10) + (3×12) = 8 + 20 + 36 = 64

Entry (2,1): Row 2 × Column 1
(4×7) + (5×9) + (6×11) = 28 + 45 + 66 = 139

Entry (2,2): Row 2 × Column 2
(4×8) + (5×10) + (6×12) = 32 + 50 + 72 = 154

Result:

[58, 64]
[139, 154]

Example 3: Real-World Style Problem

A store sells 3 products. Matrix P shows units sold this week:

P = [20, 15, 8]　Product A, B, C

Matrix C shows prices and weights:

C = [25, 2]
　　[40, 3]
　　[15, 1]

Column 1 = price per unit, Column 2 = weight per unit

What does P × C give you?

Since P is 1×3 and C is 3×2, result is 1×2.

Entry (1,1): Total revenue
(20×25) + (15×40) + (8×15) = 500 + 600 + 120 = $1,220

Entry (1,2): Total weight
(20×2) + (15×3) + (8×1) = 40 + 45 + 8 = 93 units

One multiplication, two answers. That's the power of matrix math.

Practice Problems

Problem 1

Multiply:

[3, 1] × [2, 5]
　　　[4, 6]

Problem 2

Multiply:

[1, 0, 2] × [3, 1]
　　　　　　[2, 4]
　　　　　　[1, 0]

Problem 3

If A = [2, 1] and B = [3], find A × B.
　　　　　[1, 3]　　　[1]
　　　　　　　　　　　 [2]

Problem 4

Multiply these 3×3 matrices:

[1, 2, 3] × [1, 0, 0]
[4, 5, 6]　[0, 1, 0]
[7, 8, 9]　[0, 0, 1]

Solutions

Solution 1

[3, 1] × [2, 5] = [(3×2 + 1×4), (3×5 + 1×6)] = [10, 21]
　　　[4, 6]

Solution 2

[1, 0, 2] × [3, 1] = [5, 1]
　　　　　　[2, 4]
　　　　　　[1, 0]

Check: (1×3 + 0×2 + 2×1) = 5
(1×1 + 0×4 + 2×0) = 1

Solution 3

[2, 1] × [3] = [7, 5]
[1, 3]　[1]　[6, 6]
　　　　 [2]

Entry (1,1): 2×3 + 1×1 = 7
Entry (1,2): 2×1 + 1×3 = 5
Entry (2,1): 1×3 + 3×1 = 6
Entry (2,2): 1×1 + 3×3 = 10

Solution 4

This is multiplication by the identity matrix. The result equals the original matrix:

[1, 2, 3]
[4, 5, 6]
[7, 8, 9]

Common Mistakes

When You'll Actually Use This

Computer graphics use matrix multiplication to rotate, scale, and move objects. Neural networks multiply huge matrices of weights and inputs. Game physics engines do it thousands of times per second.

The examples above are tiny. Real applications involve matrices with hundreds of rows and columns. But the process is identical. You now have the foundation.