First check the orders
(m × n)(n × p) → m × p
The inside numbers must match. The outside numbers give the order of the answer.
Learning Objectives
- understand the main ideas in Matrix Multiplication;
- use correct mathematical language and notation;
- apply the ideas to progressively harder problems;
- check and explain each solution.
Introduction
This lesson develops Matrix Multiplication from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🌱 Matrix Multiplication: Row Meets Column
Matrix multiplication is not entry-by-entry multiplication. Each answer entry is created when one row from the first matrix meets one column from the second matrix.
Select Row 1 and Column 1
2341×5216
Multiply matching terms
(2 × 5) + (3 × 1)
Multiply first, then add: 10 + 3 = 13.
Place it carefully
Row 1 × Column 1→13???
Its position is determined by the row and column used.
Repeat systematically
Row 1 × Column 1 gives entry (1,1).
Row 1 × Column 2 gives entry (1,2).
Row 2 × Column 1 gives entry (2,1).
Row 2 × Column 2 gives entry (2,2).
Identity connection
Using I returns the original matrix because each row-column calculation selects one entry and adds zeros.
AI = IA = A
Common Trap: AB and BA are usually different—and sometimes only one of them is possible.
Compatibility
For (m×n)(n×p), the inner numbers must match; the product is m×p.
(2×3)(3×4) gives 2×4. (2×3)(2×2) is impossible.
Row × Column
[2 3]
×
[4;5]
=
2×4+3×5=23
Full Example
12345678
(1,1): 1×5+2×7=19
(1,2): 1×6+2×8=22
(2,1): 3×5+4×7=43
(2,2): 3×6+4×8=50
Product=19224350.
AB is generally not equal to BA.
Practice Questions
- State the product order of (3×2)(2×5).
- Can (2×4)(3×2) be multiplied?
- Multiply 1324 by 215−1.
✅ Check Your Work
Show answers and working
- 3×5.
- No; 4≠3.
- 17−224−2.
🧮 Row-by-Column Multiplication Visualiser
Click an answer cell to see the row and column used.
1234
×5678
=Select a product entry.
📘 Remember
Key idea
AB exists when the number of columns of A equals the number of rows of B. Each product entry comes from one row times one column.
⚠️ Common Mistakes
Matrix multiplication is not entry-by-entry multiplication and generally AB ≠ BA.
⭐ Exam Tips
Write the orders first and predict the order of the answer before calculating.
🌍 Did You Know?
Matrix multiplication powers animations, 3D graphics and many machine-learning systems.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.