Multiplication of matrices
AB exists only when columns of A = rows of B. Then cᵢₖ = row i of A times column k of B, added up.
How to multiply (row by column)
Matrices are written row by row here, with rows separated by semicolons.
- 1Check the orders: A is m × n and B is n × p. The inner numbers must match, or AB is not defined.
- 2The product AB has order m × p, the outer numbers.
- 3For the entry in row i and column k, take row i of A and column k of B.
- 4★Multiply the matching entries in pairs (first with first, second with second, and so on) and add the products.
- 5Repeat for every row of A and every column of B.
★ key step
Example 1: a 2 × 3 matrix times a 3 × 2 matrix
Find AB if A = [1, 2, 0; −1, 3, 4] and B = [2, 1; 0, −1; 3, 2].
- Rule: A is 2 × 3 and B is 3 × 2, so AB exists and is 2 × 2.
- Row 1 of A with column 1 of B: (1)(2) + (2)(0) + (0)(3) = 2
- Row 1 with column 2: (1)(1) + (2)(−1) + (0)(2) = −1
- Row 2 with column 1: (−1)(2) + (3)(0) + (4)(3) = 10
- Row 2 with column 2: (−1)(1) + (3)(−1) + (4)(2) = 4
- Note: BA also exists, but it is 3 × 3, so AB and BA cannot be equal here.
AB = [2, −1; 10, 4]
Properties of matrix multiplication
| Property | Statement | Holds? |
|---|---|---|
| Associative | (AB)C = A(BC) | Yes, whenever the products are defined |
| Distributive | A(B + C) = AB + AC and (A + B)C = AC + BC | Yes |
| Multiplicative identity | IA = AI = A for a square matrix A | Yes |
| Commutative | AB = BA | No, not in general |
| Cancellation | AB = AC ⇒ B = C | No, not in general |
Example 2: a matrix polynomial (board style)
If A = [2, 1; 1, 2], show that A² − 4A + 3I = O.
- Rule: A² = A × A, multiplied row by column.
- A² = [(2)(2) + (1)(1), (2)(1) + (1)(2); (1)(2) + (2)(1), (1)(1) + (2)(2)] = [5, 4; 4, 5]
- 4A = [8, 4; 4, 8] and 3I = [3, 0; 0, 3]
- A² − 4A + 3I = [5 − 8 + 3, 4 − 4 + 0; 4 − 4 + 0, 5 − 8 + 3]
- = [0, 0; 0, 0] = O
A² − 4A + 3I = O, as required.
Example 3: a bill as a matrix product
A family buys 10 kg rice, 4 kg dal and 6 kg sugar. The prices are ₹ 50, ₹ 120 and ₹ 45 per kg (example rates). Find the bill as a matrix product.
- Rule: a 1 × 3 row matrix times a 3 × 1 column matrix gives a 1 × 1 matrix.
- Quantities Q = [10, 4, 6]; prices P = [50; 120; 45].
- QP = [(10)(50) + (4)(120) + (6)(45)]
- = [500 + 480 + 270] = [1250]
The bill is ₹ 1250.
A canteen's daily takings are a matrix product: the row of quantities sold times the column of prices. Phone graphics rotate and resize photos by multiplying the coordinates of each point by a matrix.
1 marks: “If A is 3 × 4 and B is 4 × 2, what is the order of AB? Is BA defined?” AB is 3 × 2. BA would need the number of columns of B (2) to equal the number of rows of A (3), so BA is not defined.
2 marks: “If A = [2, −1; 1, 0], show that A² − 2A + I = O.” A² = [4 − 1, −2 + 0; 2 + 0, −1 + 0] = [3, −2; 2, −1]. 2A = [4, −2; 2, 0]. So A² − 2A + I = [3 − 4 + 1, −2 + 2 + 0; 2 − 2 + 0, −1 − 0 + 1] = O.
3 marks: “Find x if [1, x] [2, 1; 3, 0] [x; 1] = [6].” [1, x][2, 1; 3, 0] = [2 + 3x, 1]. Then [2 + 3x, 1][x; 1] = [3x² + 2x + 1]. So 3x² + 2x + 1 = 6, that is 3x² + 2x − 5 = 0, or (3x + 5)(x − 1) = 0. x = 1 or x = −5/3.
Inner numbers match, outer numbers give the order: (m × n)(n × p) = m × p. Each entry: along the row, down the column, multiply and add.