1. Chapter at a Glance
A shop records the sales of three products in four branches; a computer stores an image as a grid of pixel values; an economist tracks how industries buy from each other. All of these are naturally written as rectangular arrays of numbers — matrices. Matrices let us handle many numbers at once, solve systems of equations compactly, and describe transformations in graphics. This chapter covers types of matrices, their algebra (addition, scalar and matrix multiplication), transpose, symmetric and skew-symmetric matrices, and the idea of an invertible matrix.
- Matrix notation, order m × n, elements aᵢⱼ.
- Types: row, column, square, diagonal, scalar, identity, zero.
- Equality, addition, scalar multiplication and their properties.
- Matrix multiplication; why AB ≠ BA in general.
- Transpose and its properties.
- Symmetric and skew-symmetric matrices; splitting any square matrix into these.
- Invertible matrices: definition and uniqueness of the inverse.
2. Matrices and Their Types
A matrix is a rectangular array of numbers. A matrix with m rows and n columns has order m × n; A = [aᵢⱼ] where aᵢⱼ is in row i, column j.
| Type | Description | Example |
|---|---|---|
| Row / column | 1 × n / m × 1 | [1 2 3] / [1; 2] |
| Square | m = n | 2 × 2, 3 × 3 |
| Diagonal | Square; all off-diagonal entries 0 | diag(2, 5, −1) |
| Scalar | Diagonal with equal diagonal entries | diag(4, 4, 4) |
| Identity I | Scalar with diagonal entries 1 | I₂, I₃ |
| Zero O | All entries 0 | O₂ₓ₃ |
A matrix with mn elements can have orders given by any factor pair of mn: 12 elements → 1 × 12, 2 × 6, 3 × 4, 4 × 3, 6 × 2, 12 × 1.
Construct a 2 × 2 matrix with aᵢⱼ = (i + 2j)²/2.
Solution: a₁₁ = 9/2, a₁₂ = 25/2, a₂₁ = 8, a₂₂ = 18. A = [9/2 25/2; 8 18].
3. Addition and Scalar Multiplication
- Equality: same order and equal corresponding entries.
- Addition: only for matrices of the same order; add corresponding entries.
- Scalar multiple: kA = [k aᵢⱼ].
A + B = B + A; (A + B) + C = A + (B + C); A + O = A; A + (−A) = O; k(A + B) = kA + kB; (k + l)A = kA + lA.
Find x and y if 2[1 3; 0 x] + [y 0; 1 2] = [5 6; 1 8].
Solution: 2 + y = 5 → y = 3; 2x + 2 = 8 → x = 3.
4. Matrix Multiplication
If A is m × n and B is n × p, then AB is m × p, with
The number of columns of A must equal the number of rows of B.
Find AB and BA for A = [1 2; 3 4], B = [0 1; 1 0].
Solution: AB = [2 1; 4 3] (columns swapped). BA = [3 4; 1 2] (rows swapped). AB ≠ BA.
- Associative: (AB)C = A(BC). Distributive: A(B + C) = AB + AC.
- Identity: AI = IA = A.
- Not commutative in general: AB ≠ BA.
- AB = O does not imply A = O or B = O: for example [0 1; 0 0] × [1 0; 0 0] = O, although neither factor is O.
Practice data: a school buys pens and notebooks for two classes. Class X needs 30 pens and 20 notebooks; Class Y needs 25 and 40. A pen costs ₹10 and a notebook ₹40. Find each class's cost.
Solution: [30 20; 25 40][10; 40] = [300 + 800; 250 + 1600] = [1100; 1850]: ₹1,100 and ₹1,850.
5. Transpose; Symmetric and Skew-symmetric Matrices
The transpose A′ (or Aᵀ) is obtained by interchanging rows and columns: if A = [aᵢⱼ] then A′ = [aⱼᵢ].
(A′)′ = A; (kA)′ = kA′; (A + B)′ = A′ + B′; (AB)′ = B′A′ (order reverses).
- Symmetric: A′ = A (aᵢⱼ = aⱼᵢ).
- Skew-symmetric: A′ = −A (aᵢⱼ = −aⱼᵢ; so all diagonal entries are 0).
- For any square A: A + A′ is symmetric, A − A′ is skew-symmetric.
- Every square matrix = symmetric part + skew-symmetric part: A = ½(A + A′) + ½(A − A′), uniquely.
Express A = [3 5; 1 −1] as the sum of a symmetric and a skew-symmetric matrix.
Solution: A′ = [3 1; 5 −1]. P = ½(A + A′) = [3 3; 3 −1] (symmetric). Q = ½(A − A′) = [0 2; −2 0] (skew). P + Q = A ✓.
6. Invertible Matrices
A square matrix A is invertible if there is a square matrix B of the same order with AB = BA = I. Then B = A⁻¹. If an inverse exists it is unique, and (AB)⁻¹ = B⁻¹A⁻¹.
Show that B = [3 −1; −5 2] is the inverse of A = [2 1; 5 3].
Solution: AB = [6 − 5, −2 + 2; 15 − 15, −5 + 6] = I, and BA = I similarly. ✓
(Finding inverses systematically uses determinants and adjoints — next chapter.)
7. Solved Problems (Board Pattern)
If A = [3 −2; 4 −2], find k such that A² = kA − 2I.
Solution: A² = [9 − 8, −6 + 4; 12 − 8, −8 + 4] = [1 −2; 4 −4]. kA − 2I = [3k − 2, −2k; 4k, −2k − 2]. Comparing: 3k − 2 = 1 → k = 1 (check −2k = −2 ✓, −2k − 2 = −4 ✓).
If A = [cos α −sin α; sin α cos α], show that A′A = I.
Solution: A′A = [cos²α + sin²α, 0; 0, sin²α + cos²α] = I. ∎
Verify (AB)′ = B′A′ for A = [1; −4; 3] and B = [−1 2 1].
Solution: AB = [−1 2 1; 4 −8 −4; −3 6 3]; (AB)′ = [−1 4 −3; 2 −8 6; 1 −4 3]. B′A′ = [−1; 2; 1][1 −4 3] = same matrix ✓.
8. Master Formula Table
| Result | Formula |
|---|---|
| Product order | (m × n)(n × p) = m × p |
| Product entry | (AB)ᵢₖ = Σⱼ aᵢⱼbⱼₖ |
| Transpose of product | (AB)′ = B′A′ |
| Symmetric | A′ = A |
| Skew-symmetric | A′ = −A, diagonal = 0 |
| Decomposition | A = ½(A + A′) + ½(A − A′) |
| Inverse | AA⁻¹ = A⁻¹A = I; (AB)⁻¹ = B⁻¹A⁻¹ |
9. Master Summary Table
| Operation | Condition | Key property |
|---|---|---|
| Addition | Same order | Commutative, associative |
| Scalar multiple | Any matrix | Distributive |
| Multiplication | cols(A) = rows(B) | Associative, not commutative |
| Transpose | Any matrix | Reverses products |
| Inverse | Square, invertible | Unique |
10. Mind Map
- Square, diagonal
- Identity, zero
- +, kA
- AB ≠ BA
- (AB)′ = B′A′
- Symmetric, skew
- AB = BA = I
- Unique
11. Tips and Tricks
- Before multiplying, write the orders side by side: (2 × 3)(3 × 1) → inner numbers match, outer give 2 × 1.
- To check AB = I quickly, check one product; for square matrices, AB = I implies BA = I.
- In skew-symmetric matrices, fill the diagonal with zeros immediately.
- Using (A + B)² = A² + 2AB + B² — correct only if AB = BA.
- (AB)′ = A′B′ — wrong order.
- Cancelling: AB = AC does not imply B = C.
12. Important One-liners
- Matrix multiplication is not commutative.
- Every square matrix is uniquely the sum of a symmetric and a skew-symmetric matrix.
- Diagonal elements of a skew-symmetric matrix are zero.
- (AB)′ = B′A′ and (AB)⁻¹ = B⁻¹A⁻¹.
- An inverse, if it exists, is unique.
13. Exam-pattern Questions with Detailed Answers
If A is 3 × 4 and B is 4 × 2, the order of AB is:
- 3 × 2
- 4 × 4
- 2 × 3
- not defined
Show answer
(3 × 4)(4 × 2) = 3 × 2.
Answer: (A)
How many possible orders does a matrix with 8 elements have?
Show answer
1 × 8, 2 × 4, 4 × 2, 8 × 1 → 4.
Answer: 4
If A = [aᵢⱼ] is skew-symmetric of order 3, what is a₂₂?
Show answer
Diagonal entries of a skew-symmetric matrix are 0.
Answer: 0
Compute [1 −2; 2 3][1 2 3; −3 2 −1].
Show answer
Row 1: 1 + 6, 2 − 4, 3 + 2 = 7, −2, 5. Row 2: 2 − 9, 4 + 6, 6 − 3 = −7, 10, 3.
Answer: [7 −2 5; −7 10 3]
If A = [1 0; −1 7], find k so that A² = 8A + kI.
Show answer
A² = [1 0; −8 49]. 8A = [8 0; −8 56]. 1 = 8 + k → k = −7 (check 49 = 56 − 7 ✓).
Answer: k = −7
Show that A = [0 1 −1; −1 0 1; 1 −1 0] is skew-symmetric.
Show answer
A′ = [0 −1 1; 1 0 −1; −1 1 0] = −A. ✓
Answer: Proved
Find x, y, z if [x + y + z; x + z; y + z] = [9; 5; 7].
Show answer
From first two: y = 4; then z = 3, x = 2.
Answer: x = 2, y = 4, z = 3
If A and B are symmetric matrices of the same order, show that AB − BA is skew-symmetric.
Show answer
(AB − BA)′ = B′A′ − A′B′ = BA − AB = −(AB − BA). ∎
Answer: Proved
If A = [3 1; −1 2], show that A² − 5A + 7I = O.
Show answer
A² = [8 5; −5 3]. 5A = [15 5; −5 10]. A² − 5A + 7I = [8 − 15 + 7, 0; 0, 3 − 10 + 7] = O. ∎
Answer: Proved
Express [6 −2 2; −2 3 −1; 2 −1 3] as the sum of a symmetric and a skew-symmetric matrix.
Show answer
A is already symmetric (A′ = A), so P = A and Q = O.
Answer: A = A + O
If A = [0 −tan(α/2); tan(α/2) 0] and I is the 2 × 2 identity, show that I + A = (I − A)[cos α −sin α; sin α cos α].
Show answer
Let t = tan(α/2): cos α = (1 − t²)/(1 + t²), sin α = 2t/(1 + t²). (I − A) = [1 t; −t 1]. Product row 1: [cos α + t sin α, −sin α + t cos α] = [(1 − t² + 2t²)/(1 + t²), (−2t + t − t³)/(1 + t²)] = [1, −t]. Row 2 similarly gives [t, 1]. So the product = [1 −t; t 1] = I + A. ∎
Answer: Proved
Practice data: a trust invests ₹30,000 in two bonds paying 5% and 7% interest. Using matrix multiplication, find how to divide the money to earn ₹1,800 a year.
Show answer
[x 30000 − x][0.05; 0.07] = 1800 → 0.05x + 2100 − 0.07x = 1800 → 0.02x = 300 → x = 15000. Invest ₹15,000 in each.
Answer: ₹15,000 and ₹15,000
14. Quick Revision Summary
- Order m × n; types: square, diagonal, scalar, identity, zero.
- Add same orders; multiply when columns of A = rows of B; AB ≠ BA generally.
- Transpose: (AB)′ = B′A′.
- Symmetric A′ = A; skew A′ = −A; A = ½(A + A′) + ½(A − A′).
- Inverse: AB = BA = I; unique; (AB)⁻¹ = B⁻¹A⁻¹.