All subjects · Further Mathematics · WFM01 Further Pure Mathematics 1

Matrix arithmetic

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

A matrix is a rectangular array of numbers. In F1 you add, subtract and multiply matrices and multiply them by numbers. Matrix multiplication has rules of its own: in particular, order matters.

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Size, adding and scalars

A matrix with m rows and n columns has order m × n. Matrices of the same order are added or subtracted entry by entry. A scalar multiple kA multiplies every entry by k.

The zero matrix 0 has every entry 0; the identity I = (1 0; 0 1) has 1s on the leading diagonal. In this site (a b; c d) means the 2 × 2 matrix with top row a b and bottom row c d.

Multiplying

AB exists only when the number of columns of A equals the number of rows of B. An m × n matrix times an n × p matrix is m × p.

Each entry of AB is a row of A times a column of B: multiply matching entries and add.

(2 -1; 3 4)(1 5; 0 -2) = (2×1 + (-1)×0, 2×5 + (-1)(-2); 3×1 + 4×0, 3×5 + 4(-2)) = (2 12; 3 7).

Rules

In general AB ≠ BA.
Multiplication is associative: (AB)C = A(BC).
AI = IA = A.
A² means AA, which needs A to be square.
Because AB ≠ BA, (A + B)² = A² + AB + BA + B², and A² - B² is not usually (A - B)(A + B).

Worked example

A = (2 -1; 3 4) and B = (1 5; 0 -2). Find AB and BA.

AB = (2 12; 3 7)
BA: row 1 is (1×2 + 5×3, 1×(-1) + 5×4) = (17, 19); row 2 is (0×2 + (-2)×3, 0×(-1) + (-2)×4) = (-6, -8).

Answer: AB = (2 12; 3 7), BA = (17 19; -6 -8): different

Key idea: Add entry by entry; multiply row by column. AB needs columns of A = rows of B, and usually AB ≠ BA.

Check yourself

A = (2 -1; 3 4) and B = (1 5; 0 -2). Find the bottom-right entry of AB.
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7. Row 2 of A times column 2 of B: 3×5 + 4×(-2) = 15 - 8 = 7.
A = (2 -1; 3 4) and B = (1 5; 0 -2). Find the top-left entry of BA.
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17. Row 1 of B times column 1 of A: 1×2 + 5×3 = 17.
A = (2 -1; 3 4) and B = (1 5; 0 -2). Find the top-right entry of 3A - 2B.
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-13. 3×(-1) - 2×5 = -3 - 10 = -13.
A = (2 -1; 3 4). Find the bottom-left entry of A².
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18. A² = AA. Row 2 of A times column 1 of A: 3×2 + 4×3 = 18.
C is 2 × 3 and D is 3 × 2. What is the order of DC?
- 2 × 2
- 3 × 2
- 3 × 3
- It does not exist
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3 × 3. D is 3 × 2 and C is 2 × 3: the inner numbers match (2 and 2), and the result takes the outer numbers, 3 × 3.
C = (1 0 2; -1 3 1) and D = (2 1; 0 4; 1 -1). Find the top-left entry of CD.
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4. Row 1 of C times column 1 of D: 1×2 + 0×0 + 2×1 = 4.
A and B are 2 × 2 matrices. Which statement is always true?
- AB = BA
- (A + B)² = A² + 2AB + B²
- (AB)C = A(BC)
- If AB = 0 then A = 0 or B = 0
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(AB)C = A(BC). Matrix multiplication is associative but not commutative.

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Last updated 8 November 2026. Spotted a mistake? Tell us.