GCSE Further Maths · Topic 4
Matrices
Matrices are grids of numbers that can move shapes around.
We will multiply matrices and use the identity matrix.
Then we see how a matrix reflects, rotates or enlarges the unit square, and how to combine transformations.
- 2 subtopics
- 31 questions
- Free: 4.1 notes and 10 questions
In this topic
- 4.1Matrices: multiplying matrices and the identity matrixFree below · 16 questions
- 4.2Matrix transformations of the unit square: reflections, rotations, enlargements and combinationsOpen in Full Marks · 15 questions
4.1 · Free notes
Matrices: multiplying matrices and the identity matrix
Key points
- A matrix is a rectangular array of numbers; its order is rows × columns. A 2 × 2 matrix has 2 rows and 2 columns.
- Multiplying: each entry of AB is a row of A times a column of B (multiply pairs and add). [[a, b], [c, d]] × [[e], [f]] = [[ae + bf], [ce + df]].
- Matrix multiplication is not commutative: usually AB ≠ BA. You can only multiply when the number of columns of A equals the number of rows of B.
- The identity matrix I = [[1, 0], [0, 1]] leaves any matrix unchanged: AI = IA = A.
- Multiplying by a number (scalar) multiplies every entry: 3[[1, 2], [0, −1]] = [[3, 6], [0, −3]].
In more detail
- Worked example: if [[2, a], [1, 3]] × [[1], [2]] = [[8], [7]], then 2 + 2a = 8 gives a = 3 (check: 1 + 6 = 7).
- Why order matters: applying transformation B then A is the matrix AB, which is generally different from BA.
- Exam tip: write out each entry calculation (row × column) to avoid arithmetic slips.
Practice
Matrices: multiplying matrices and the identity matrix: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. What is the order of a matrix with 2 rows and 3 columns?
- 3 × 2
- 6
- 5
- 2 × 3
Show answer
D: 2 × 3. Rows × columns.
2. Calculate [[2, 1], [0, 3]] × [[4], [5]].
- [[8], [15]]
- [[13], [15]]
- [[9], [15]]
- [[13], [5]]
Show answer
B: [[13], [15]]. Row 1: 8 + 5 = 13; row 2: 0 + 15 = 15.
3. Calculate [[1, 2], [3, 4]] × [[1, 0], [0, 1]].
- [[1, 2], [3, 4]]
- [[1, 0], [0, 4]]
- [[2, 2], [3, 5]]
- [[1, 2], [0, 4]]
Show answer
A: [[1, 2], [3, 4]]. Multiplying by the identity leaves the matrix unchanged.
4. Is matrix multiplication commutative?
- Yes, always
- Only for 3 × 3 matrices
- No, AB is usually not equal to BA
- Only for vectors
Show answer
C: No, AB is usually not equal to BA. Order matters.
5. Calculate [[1, 2], [0, 1]] × [[3, 0], [1, 2]].
- [[3, 2], [1, 2]]
- [[3, 4], [1, 2]]
- [[5, 2], [1, 3]]
- [[5, 4], [1, 2]]
Show answer
D: [[5, 4], [1, 2]]. Row 1: (3 + 2, 0 + 4); row 2: (1, 2).
6. What is 2 × [[3, −1], [0, 4]]?
- [[5, 1], [2, 6]]
- [[6, −2], [0, 8]]
- [[6, −1], [0, 4]]
- [[3, −2], [0, 8]]
Show answer
B: [[6, −2], [0, 8]]. Multiply every entry by 2.
7. What is the 2 × 2 identity matrix?
- [[1, 0], [0, 1]]
- [[1, 1], [1, 1]]
- [[0, 1], [1, 0]]
- [[0, 0], [0, 0]]
Show answer
A: [[1, 0], [0, 1]]. 1s on the leading diagonal, 0s elsewhere.
8. Can a 2 × 3 matrix be multiplied by a 2 × 2 matrix (in that order)?
- Yes
- Only if square
- No
- Only if the entries are integers
Show answer
C: No. Columns of the first (3) must equal rows of the second (2).
9. If A = [[2, 0], [0, 2]], what is A²?
- [[2, 0], [0, 2]]
- [[4, 4], [4, 4]]
- [[0, 4], [4, 0]]
- [[4, 0], [0, 4]]
Show answer
D: [[4, 0], [0, 4]]. A × A.
10. Calculate [[0, −1], [1, 0]] × [[2], [3]].
- [[3], [−2]]
- [[−3], [2]]
- [[2], [3]]
- [[−2], [3]]
Show answer
B: [[−3], [2]]. Row 1: 0 − 3 = −3; row 2: 2 + 0 = 2.
Keep going in Full Marks
Full Marks has 16 questions on Matrices: multiplying matrices and the identity matrix (6 more than here), set out in practice levels. You also get notes and 3D models for all 2 subtopics of Matrices, plus games, timed exams and past papers.
On your exam board
Not in the OCR A and Cambridge specifications.