Compare with algebra
For 3x = 12, multiply both sides by 1/3.
For AX = B, multiply by A⁻¹.
Learning Objectives
- understand the main ideas in Matrix Equations;
- use correct mathematical language and notation;
- apply the ideas to progressively harder problems;
- check and explain each solution.
Introduction
This lesson develops Matrix Equations 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.
🌱 Solving Equations with Matrices
A matrix equation follows the same idea as an ordinary equation: undo the operation while keeping the order of multiplication correct.
Order matters
Start: AX = B.
Left-multiply both sides by A⁻¹.
A⁻¹AX = A⁻¹B.
IX = A⁻¹B, so X = A⁻¹B.
Why the identity disappears
Because IX = X, just as 1x = x.
X = A⁻¹B
Do not reverse the product
A⁻¹B and BA⁻¹ are generally different. Preserve the side on which A acts.
Before Solving: Check that A is square and det(A) ≠ 0, otherwise A⁻¹ does not exist.
AX=B
Multiply on the left by A⁻¹: X=A⁻¹B.
Order matters; X=BA⁻¹ is not generally equivalent.
Example
A=2111, B=74, A⁻¹=1−1−12.
X=A⁻¹B=31.
Check AX=74.
Practice Questions
- Solve AX=B for A=1123, B=513.
- Why must multiplication order be preserved?
✅ Check Your Work
Show answers and working
- A⁻¹=3−1−21, so X=23.
- Matrix multiplication is not commutative.
🧠 Matrix Equation Solver
Solve AX=B for the 2×1 matrix X.
X=
📘 Remember
Key idea
To solve AX=B, multiply both sides on the left by A⁻¹, giving X=A⁻¹B.
⚠️ Common Mistakes
Matrix multiplication order matters. You cannot casually move a matrix across an equation like an ordinary number.
⭐ Exam Tips
Check that A is invertible before attempting to solve.
🌍 Did You Know?
Matrix equations compactly represent many linked equations at once.
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.