Class 12 Maths Notes: Matrices (Chapter 3)
Master Chapter 3 of Class 12 Mathematics. Understand matrix multiplication rules, transpose properties, symmetric and skew-symmetric matrices, inverse, and common exam errors with PYQs.
Students assume matrix operations work like ordinary number operations. They try to treat AB = BA, cancel matrix factors, and divide by matrices. These assumptions do not hold in matrix algebra and produce consistent, avoidable errors.
1. Matrix Algebra Is Not Ordinary Algebra
| Ordinary Algebra (True) | Matrix Algebra (False!) |
|---|---|
| ab = ba always | AB ≠ BA in general |
| If ab = 0, then a = 0 or b = 0 | AB = O does not mean A = O or B = O |
| If ab = ac and a ≠ 0, then b = c | AB = AC does not mean B = C |
2. Matrix Multiplication: Compatibility Rule
Matrix AB is defined only when the number of columns in A equals the number of rows in B.
Order Rule
If A is m × n and B is n × p, then AB is m × p.
Always verify compatibility before multiplying. If A is 2×3 and B is 2×3, then AB is not defined. But note: even if AB is defined, BA may not be.
3. Transpose Properties
- (A + B)ᵀ = Aᵀ + Bᵀ — same order (no reversal)
- (AB)ᵀ = BᵀAᵀ — order reverses!
- (Aᵀ)ᵀ = A
The rule: transpose of a product reverses the order. Transpose of a sum does not. Students who apply the same rule to both make sign errors in proofs.
4. Symmetric and Skew-Symmetric Matrices
| Type | Condition | Diagonal Elements |
|---|---|---|
| Symmetric | A = Aᵀ (i.e., aᵢⱼ = aⱼᵢ) | Can be any value |
| Skew-Symmetric | A = −Aᵀ (i.e., aᵢⱼ = −aⱼᵢ) | Must all be zero |
Standard board proof: every square matrix A can be expressed as A = ½(A + Aᵀ) + ½(A − Aᵀ), where the first term is symmetric and the second is skew-symmetric.
5. Inverse of a Matrix
Formula: Inverse Using Adjugate
A⁻¹ = adj(A) / |A|
Step 1: Calculate |A|. If |A| = 0, the inverse does not exist (singular matrix — stop here).
Step 2: Find the matrix of cofactors.
Step 3: Transpose the cofactor matrix to get adj(A). Students who forget to transpose get the wrong inverse.
⚠️ Watch Out! — Matrix Equation Side
To solve AX = B: multiply both sides by A⁻¹ on the left → X = A⁻¹B.
Writing X = BA⁻¹ solves the different equation XA = B. Since matrix multiplication is not commutative, these are different answers. Always multiply on the correct side.
Summary: Key Rules at a Glance
| Concept | Key Rule |
|---|---|
| Multiplication commutativity | AB ≠ BA in general |
| Transpose of product | (AB)ᵀ = BᵀAᵀ (order reverses) |
| Skew-symmetric diagonal | All diagonal elements = 0 |
| Inverse exists when | |A| ≠ 0 |
| Solving AX = B | X = A⁻¹B (multiply on the left) |
Practice Questions (PYQs)
- If A is a square matrix, show that A + Aᵀ is always symmetric and A − Aᵀ is always skew-symmetric.
- Find the inverse of the matrix A = [[2, 1], [5, 3]] using the adjugate method. Verify that AA⁻¹ = I.
- Give an example of two non-zero matrices A and B such that AB = O (zero matrix). What does this tell you about matrix cancellation?
- If (AB)ᵀ = BᵀAᵀ, prove this property for 2×2 matrices using a specific example.
- Solve the matrix equation AX = B where A = [[1, 2], [3, 4]] and B = [[5], [6]] for X.