Matrices
Introduction
A matrix is a rectangular arrangement of numbers arranged in rows and columns. In ICSE Class 10, you learn the basic operations on matrices — addition, subtraction, scalar multiplication, and matrix multiplication.
Notation
A matrix is denoted by a capital letter, e.g., A. The element in the i-th row and j-th column is denoted by aᵢⱼ.
A = [aᵢⱼ]ₘₓₙ means a matrix with m rows and n columns.
Order of a matrix = number of rows × number of columns (m × n).
Example: A 2 × 3 matrix:
A = [1 2 3]
[4 5 6]
Types of Matrices
| Type | Definition | Example |
|---|---|---|
| Row matrix | Only 1 row (1 × n) | [2 3 5] |
| Column matrix | Only 1 column (m × 1) | [2; 3; 5] |
| Square matrix | Equal rows and columns (m = n) | [1 2; 3 4] |
| Zero matrix | All elements are 0 | [0 0; 0 0] |
| Identity matrix | Square with 1s on diagonal, 0s elsewhere | [1 0; 0 1] |
| Diagonal matrix | Non-zero elements only on diagonal | [3 0; 0 5] |
| Scalar matrix | Diagonal with all equal elements | [4 0; 0 4] |
| Symmetric matrix | Aᵀ = A (aᵢⱼ = aⱼᵢ) | [1 2; 2 3] |
Matrix Operations
Equality of Matrices
Two matrices are equal if they have the same order and corresponding elements are equal.
A = B ⇔ aᵢⱼ = bᵢⱼ for all i, j
Addition and Subtraction
Matrices of the same order can be added/subtracted by adding/subtracting corresponding elements.
(A ± B)ᵢⱼ = aᵢⱼ ± bᵢⱼ
Example:
A = [1 2] B = [3 4]
[5 6] [7 8]
A + B = [4 6]
[12 14]
A − B = [-2 -2]
[-2 -2]
Scalar Multiplication
Each element is multiplied by the scalar.
(kA)ᵢⱼ = k × aᵢⱼ
Matrix Multiplication
Two matrices can be multiplied only if the number of columns in A equals the number of rows in B.
Aₘₓₙ × Bₙₓₚ = Cₘₓₚ
The element cᵢⱼ = sum of products of i-th row of A and j-th column of B.
Example: Multiply A₂ₓ₂ and B₂ₓ₂
A = [1 2] B = [5 6]
[3 4] [7 8]
AB = [1×5+2×7 1×6+2×8] = [19 22]
[3×5+4×7 3×6+4×8] [43 50]
Important: Matrix multiplication is NOT commutative. AB ≠ BA generally.
Transpose of a Matrix
The transpose of A (denoted Aᵀ or A') is obtained by interchanging rows and columns.
If A = [aᵢⱼ]ₘₓₙ, then Aᵀ = [aⱼᵢ]ₙₓₘ
Properties of transpose:
- (Aᵀ)ᵀ = A
- (A + B)ᵀ = Aᵀ + Bᵀ
- (kA)ᵀ = kAᵀ
- (AB)ᵀ = BᵀAᵀ
Worked Examples
Example 1: Finding Unknown Elements
If [x+y 3; 5 y−z] = [7 3; 5 4], find x, y, z.
Solution: Equating corresponding elements: x + y = 7 ...(i) 3 = 3 ✓ 5 = 5 ✓ y − z = 4 ...(ii)
From (i): if y = 3, x = 4 From (ii): 3 − z = 4 → z = −1
x = 4, y = 3, z = −1
Example 2: Matrix Multiplication
If A = [2 −1; 0 3] and B = [1 4; −2 5], find AB and BA. Show that AB ≠ BA.
Solution: AB = [2×1+(−1)×(−2) 2×4+(−1)×5; 0×1+3×(−2) 0×4+3×5] = [2+2 8−5; 0−6 0+15] = [4 3; −6 15]
BA = [1×2+4×0 1×(−1)+4×3; (−2)×2+5×0 (−2)×(−1)+5×3] = [2+0 −1+12; −4+0 2+15] = [2 11; −4 17]
Since AB ≠ BA, matrix multiplication is not commutative. ✓
Example 3: Transpose and Multiplication
Given A = [1 2; 3 4], verify that (Aᵀ)ᵀ = A.
Solution: Aᵀ = [1 3; 2 4] (Aᵀ)ᵀ = [1 2; 3 4] = A ✓
Comparison: Matrix Operations
| Operation | Condition on Order | Result Order |
|---|---|---|
| Addition/Subtraction | Same order | Same order |
| Scalar multiplication | Any order | Same order |
| Multiplication (AB) | Columns(A) = Rows(B) | Rows(A) × Columns(B) |
| Transpose | Any order | n × m |
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| Adding matrices of different orders | Only same-order matrices can be added |
| Multiplying in wrong order | AB ≠ BA — check dimensions carefully |
| Incorrect element pairing in multiplication | Multiply row of A with column of B |
| Mixing up rows and columns in order | Order = rows × columns |
| Forgetting the transpose swaps rows/columns | Aᵢⱼ becomes Aⱼᵢ in transpose |
ICSE Exam Focus
Matrices carry 8–10 marks in ICSE exams. Questions include:
- Finding unknowns using equality of matrices.
- Matrix addition, subtraction, and scalar multiplication.
- Matrix multiplication (2×2 and 2×3 with 3×2).
- Proving non-commutativity of multiplication.
- Transpose operations.
Marks Blueprint:
| Topic | Marks |
|---|---|
| Equality of matrices (finding unknowns) | 3 |
| Addition/Subtraction/Scalar multiplication | 3 |
| Matrix multiplication | 4 |
| Transpose and properties | 2 |
Self-Test Questions
-
If [2x + y 4; 3 x − 2y] = [7 4; 3 2], find x and y.
-
Given A = [3 1; −2 4] and B = [1 2; 3 −1], find 2A − 3B.
-
If A = [2 3; 1 −4] and B = [1 −2; −1 3], find AB and BA. Show that AB ≠ BA.
-
If A = [1 2 3; 4 5 6]₂ₓ₃ and B = [7 8; 9 10; 11 12]₃ₓ₂, find AB.
-
Define symmetric matrix. If A = [2 3; 3 1], verify that A is symmetric by showing A = Aᵀ.
In ICSE, matrix multiplication questions frequently include verification of non-commutativity. Always show both AB and BA explicitly to confirm.
