A-level Further Maths Year 12 · Topic 2
Matrices
Matrices describe transformations and solve systems of equations.
We will multiply matrices, find determinants and inverses, and see what they do to shapes in two and three dimensions.
Then we find invariant points and lines and interpret three planes.
- 5 subtopics
- 68 questions
- Free: 2.1 notes and 10 questions
In this topic
- 2.1Matrix arithmetic: addition, multiplication and special matricesFree below · 14 questions
- 2.2Determinants and inverse matrices (2 × 2 and 3 × 3)Open in Full Marks · 15 questions
- 2.3Matrix transformations in two and three dimensionsOpen in Full Marks · 13 questions
- 2.4Invariant points and invariant linesOpen in Full Marks · 13 questions
- 2.5Systems of linear equations and the geometry of three planesOpen in Full Marks · 13 questions
2.1 · Free notes
Matrix arithmetic: addition, multiplication and special matrices
Key points
- Matrices of the same order can be added or subtracted entry by entry. Multiply by a scalar by multiplying every entry.
- Matrix multiplication AB needs columns of A = rows of B; an m × n matrix times an n × p matrix gives m × p. Each entry is a row of A times a column of B.
- Matrix multiplication is associative ((AB)C = A(BC)) but not commutative (AB ≠ BA in general).
- The identity I has 1s on the leading diagonal and 0s elsewhere: AI = IA = A. The zero matrix has all entries 0.
- Powers: A² = AA, A³ = AAA. Some matrices satisfy special properties such as A² = I (self-inverse).
In more detail
- Worked example: find k if [[1, k], [0, 1]]² = [[1, 6], [0, 1]]: the top-right entry is 2k, so k = 3.
- Proving AB ≠ BA: one counter-example with specific matrices is enough.
- Exam tip: on a calculator, check matrix products, but show at least one entry's working in the exam.
Practice
Matrix arithmetic: addition, multiplication and special matrices: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. What is the order of AB if A is 2 × 3 and B is 3 × 4?
- 3 × 3
- 4 × 2
- Not possible
- 2 × 4
Show answer
D: 2 × 4. Rows of A × columns of B.
2. Can a 3 × 2 matrix be multiplied by a 3 × 2 matrix?
- Yes
- No
- Only if they are equal
- Only if both are square
Show answer
B: No. Columns (2) ≠ rows (3).
3. Calculate [[1, 2], [3, 4]] + [[0, −1], [5, 2]].
- [[1, 1], [8, 6]]
- [[1, 3], [−2, 2]]
- [[0, −2], [15, 8]]
- [[1, 1], [8, 2]]
Show answer
A: [[1, 1], [8, 6]]. Add corresponding entries.
4. Calculate [[1, 2], [3, 4]] × [[2, 0], [1, 1]].
- [[2, 0], [3, 4]]
- [[2, 4], [7, 4]]
- [[4, 2], [10, 4]]
- [[4, 2], [7, 6]]
Show answer
C: [[4, 2], [10, 4]]. Row 1: (2 + 2, 0 + 2); row 2: (6 + 4, 0 + 4).
5. Which property does matrix multiplication have?
- Commutative
- Both
- Neither
- Associative
Show answer
D: Associative. (AB)C = A(BC).
6. If A = [[0, 1], [1, 0]], what is A²?
- A
- I
- 0
- 2A
Show answer
B: I. Reflecting twice returns to the start.
7. What is 3[[2, −1], [0, 4]]?
- [[6, −3], [0, 12]]
- [[5, 2], [3, 7]]
- [[6, −1], [0, 4]]
- [[2, −3], [0, 12]]
Show answer
A: [[6, −3], [0, 12]]. Multiply each entry.
8. For a matrix A, what is AI?
- I
- 0
- A
- A²
Show answer
C: A. The identity leaves A unchanged.
9. If A is 3 × 3, what is the order of A²?
- 9 × 9
- 6 × 6
- 3 × 1
- 3 × 3
Show answer
D: 3 × 3. Square matrices can be squared.
10. Calculate [[2, 1], [0, 1]]².
- [[4, 1], [0, 1]]
- [[4, 3], [0, 1]]
- [[4, 2], [0, 1]]
- [[2, 2], [0, 2]]
Show answer
B: [[4, 3], [0, 1]]. Row 1: (4 + 0, 2 + 1).
Keep going in Full Marks
Full Marks has 14 questions on Matrix arithmetic: addition, multiplication and special matrices (4 more than here), set out in practice levels. You also get notes and 3D models for all 5 subtopics of Matrices, plus games, timed exams and past papers.