Matrices
1. Introduction
A matrix is a rectangular arrangement of numbers in rows and columns. Matrices are fundamental tools in linear algebra with applications in physics, economics, computer graphics, and engineering.
2. Types of Matrices
Row Matrix: 1 × n matrix. Column Matrix: m × 1 matrix. Square Matrix: m = n. Diagonal Matrix: All non-diagonal elements are zero. Scalar Matrix: Diagonal matrix with equal diagonal elements. Identity Matrix: Diagonal matrix with diagonal entries = 1. Denoted I. Zero Matrix: All entries are zero. Denoted O. Upper Triangular Matrix: All entries below the main diagonal are zero. Lower Triangular Matrix: All entries above the main diagonal are zero.
3. Algebra of Matrices
3.1 Addition and Subtraction
Matrices of the same order can be added component-wise: (A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ.
Properties: Commutative (A + B = B + A), Associative ((A + B) + C = A + (B + C)), Additive identity O, Additive inverse -A.
3.2 Scalar Multiplication
kA is obtained by multiplying each element of A by k.
3.3 Matrix Multiplication
If A is m × n and B is n × p, then AB is m × p with (AB)ᵢⱼ = Σ aᵢₖ bₖⱼ for k = 1 to n.
'Matrix multiplication is NOT commutative. AB ≠ BA in general.' Properties: Associative, Distributive over addition. No cancellation law in general.
4. Transpose of a Matrix
(Aᵀ)ᵢⱼ = Aⱼᵢ Properties: (Aᵀ)ᵀ = A, (A + B)ᵀ = Aᵀ + Bᵀ, (kA)ᵀ = kAᵀ, (AB)ᵀ = BᵀAᵀ.
5. Symmetric and Skew-Symmetric Matrices
Symmetric: Aᵀ = A. Entries symmetric about diagonal: aᵢⱼ = aⱼᵢ. Skew-Symmetric: Aᵀ = -A. Diagonal entries are zero. aᵢⱼ = -aⱼᵢ.
Every square matrix can be expressed uniquely as: A = P + Q, where P = (A + Aᵀ)/2 is symmetric and Q = (A - Aᵀ)/2 is skew-symmetric.
6. Elementary Row and Column Operations
Three types: (i) Rᵢ ↔ Rⱼ (interchange), (ii) Rᵢ → kRᵢ (scalar multiplication), (iii) Rᵢ → Rᵢ + kRⱼ (addition).
7. Inverse Using Elementary Operations
For a square matrix A, if we can transform [A|I] to [I|A^{-1}] using elementary row operations, then A is invertible.
Algorithm: Apply row operations on A and I simultaneously until A becomes I. The resulting I becomes A^{-1}.
'If during the process a row becomes all zeros, the matrix is singular and has no inverse.'
8. Worked Problems
Problem 1: If A = [[2, 3], [1, -4]], find A², 3A + 2I, and verify that A satisfies its characteristic equation. Solution: A² = [[2, 3], [1, -4]] × [[2, 3], [1, -4]] = [[7, -6], [-2, 19]]. 3A + 2I = [[6+2, 9], [3, -12+2]] = [[8, 9], [3, -10]].
Problem 2: Find the inverse of A = [[2, 3], [1, 2]] using elementary row operations. Solution: [A|I] = [[2, 3, 1, 0], [1, 2, 0, 1]] R₁ → R₁ - R₂: [[1, 1, 1, -1], [1, 2, 0, 1]] R₂ → R₂ - R₁: [[1, 1, 1, -1], [0, 1, -1, 2]] R₁ → R₁ - R₂: [[1, 0, 2, -3], [0, 1, -1, 2]] Hence A^{-1} = [[2, -3], [-1, 2]].
Problem 3: Express A = [[1, 4], [6, 3]] as sum of symmetric and skew-symmetric matrices. Solution: Aᵀ = [[1, 6], [4, 3]]. P = (A + Aᵀ)/2 = [[1, 5], [5, 3]]. Q = (A - Aᵀ)/2 = [[0, -1], [1, 0]].
9. Common Mistakes
'Many students multiply matrices incorrectly. Remember: rows of first × columns of second. The inner dimensions must match.'
'When finding inverse by elementary operations, use ONLY row operations or ONLY column operations — never mix them.'
10. ISC Exam Focus
| Topic | Theory Marks | Practical Marks |
|---|---|---|
| Matrix algebra | 3 | 2 |
| Transpose and symmetry | 2 | 1 |
| Inverse by elementary ops | 4 | 3 |
| Solving using matrices | 4 | 2 |
11. Self-Test Questions
- If A = [[3, -2], [4, -2]], find A² - 5A + 2I.
- Find the inverse of A = [[3, 1, 2], [2, 1, 1], [1, 2, 1]] using elementary row operations.
- Express [[2, 3, -1], [4, 1, 0], [5, -2, 3]] as sum of symmetric and skew-symmetric matrices.
- If A = [[2, 3], [1, 5]], find the matrix X such that 2A + 3X = 5I.
- Prove that (A + B)² = A² + AB + BA + B² for matrices.
12. Key Formulae
| Concept | Formula |
|---|---|
| Matrix multiplication | (AB)ᵢⱼ = Σ aᵢₖbₖⱼ |
| Transpose | (AB)ᵀ = BᵀAᵀ |
| Symmetric decomposition | A = (A + Aᵀ)/2 + (A - Aᵀ)/2 |
| Inverse (elementary ops) | Apply row ops to [A |
| Distribution | A(B + C) = AB + AC |
| Transpose of sum | (A + B)ᵀ = Aᵀ + Bᵀ |
13. Additional Practice Problems
Problem A: If A = [[1, 2, 3], [2, 3, 1], [3, 1, 2]], verify that A³ - 6A² + 9A - 4I = 0.
Solution: Compute A², then A³, substitute into the polynomial and simplify. Each entry should be zero, verifying the matrix satisfies its characteristic equation.
Problem B: Find the matrix X if 2A + 3X = 5B where A = [[1, 2], [3, -1]] and B = [[2, 0], [1, 4]].
Solution: 3X = 5B - 2A = [[10, 0], [5, 20]] - [[2, 4], [6, -2]] = [[8, -4], [-1, 22]]. So X = (1/3)[[8, -4], [-1, 22]] = [[8/3, -4/3], [-1/3, 22/3]].
Problem C: If A = [[2, -1], [1, 3]], show that (A - 3I)(A + 2I) = A² - A - 6I.
Solution: A - 3I = [[-1, -1], [1, 0]], A + 2I = [[4, -1], [1, 5]]. Multiply: [[-1, -1], [1, 0]] × [[4, -1], [1, 5]] = [[-5, -4], [4, -1]] = A² - A - 6I. Verify A² = [[3, -5], [5, 14]], so A² - A - 6I = [[3-2-6, -5+1], [5-1, 14-3-6]] = [[-5, -4], [4, 5]]. The two match, proving the identity.
