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Matrices · Form 5

Matrices: Practice Questions

Original SPM-style practice questions for Matrices, each with full worked solutions, Paper 1 multiple-choice and Paper 2 structured.

Written by the spmmath.com.my editorial team.

Original practice questions for Matrices, in the style of Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective). Each answer is worked so you can check your method, not just the result.

Multiple-choice (Paper 1 style)

Question 1

Matrix P = [ 1 4 -2 ; 0 3 5 ]. What is the order of P?

  1. A. 2 × 3
  2. B. 3 × 2
  3. C. 6
  4. D. 5

Question 2

Evaluate [ 6 -1 ; 3 5 ] − [ 2 4 ; -1 2 ].

  1. A. [ 4 -5 ; 4 3 ]
  2. B. [ 4 3 ; 2 7 ]
  3. C. [ 8 3 ; 2 7 ]
  4. D. [ 4 -5 ; 2 3 ]

Question 3

Given M = [ 3 -1 ; 0 4 ], find −2M.

  1. A. [ -6 2 ; 0 -8 ]
  2. B. [ -6 -2 ; 0 -8 ]
  3. C. [ 6 -2 ; 0 8 ]
  4. D. [ -6 2 ; -2 -8 ]

Question 4

Evaluate [ 2 1 ; 0 3 ] × [ 1 -2 ; 4 1 ].

  1. A. [ 6 -3 ; 12 3 ]
  2. B. [ 2 -2 ; 0 3 ]
  3. C. [ 6 3 ; 12 -3 ]
  4. D. [ 3 -1 ; 4 4 ]

Question 5

Find the determinant of [ 5 3 ; 2 4 ].

  1. A. 14
  2. B. 26
  3. C. −14
  4. D. 16

Question 6

The inverse of [ 4 3 ; 1 1 ] is:

  1. A. [ 1 -3 ; -1 4 ]
  2. B. [ 4 -3 ; -1 1 ]
  3. C. [ 1 3 ; 1 4 ]
  4. D. [ -1 3 ; 1 -4 ]

Question 7

The matrix [ 3 k ; 2 4 ] has no inverse. Find the value of k.

  1. A. 6
  2. B. 8
  3. C. −6
  4. D. 2

Question 8

Matrix P has order 3 × 2 and matrix Q has order 2 × 5. What is the order of the product PQ?

  1. A. 3 × 5
  2. B. 5 × 3
  3. C. 2 × 2
  4. D. Not defined

Question 9

If [ x+1 3 ; 2 y−2 ] = [ 4 3 ; 2 5 ], find the value of x + y.

  1. A. 10
  2. B. 8
  3. C. 6
  4. D. 12

Question 10

The simultaneous equations 2x + 3y = 12 and x − y = 1 can be written in matrix form as:

  1. A. [ 2 3 ; 1 -1 ][ x ; y ] = [ 12 ; 1 ]
  2. B. [ 2 1 ; 3 -1 ][ x ; y ] = [ 12 ; 1 ]
  3. C. [ 2 3 ; 1 -1 ][ 12 ; 1 ] = [ x ; y ]
  4. D. [ x y ][ 2 3 ; 1 -1 ] = [ 12 ; 1 ]

Structured (Paper 2 style)

Question 1 (4 marks)

Given P = [ 1 0 ; 2 -3 ] and Q = [ 3 4 ; -1 5 ]. (a) Find the matrix 2P + Q.

(b) Find the determinant of Q.

  1. (a) Scalar multiply P by 2: 2P = [ 2 0 ; 4 -6 ].
  2. Add corresponding elements of 2P and Q: 2+3=5, 0+4=4, 4+(−1)=3, −6+5=−1.
  3. So 2P + Q = [ 5 4 ; 3 -1 ].
  4. (b) For Q = [ a b ; c d ], determinant = ad − bc = (3×5) − (4×(−1)) = 15 + 4 = 19.

Question 2 (4 marks)

Given M = [ 3 2 ; 5 4 ]. (a) Find the determinant of M.

(b) Hence find the inverse matrix M⁻¹.

  1. (a) Determinant = ad − bc = (3×4) − (2×5) = 12 − 10 = 2.
  2. (b) The inverse of [ a b ; c d ] is (1÷determinant)[ d -b ; -c a ].
  3. Swap a and d, negate b and c: [ 4 -2 ; -5 3 ].
  4. Multiply by 1÷2: M⁻¹ = ½[ 4 -2 ; -5 3 ] = [ 2 -1 ; -2.5 1.5 ].

Question 3 (5 marks)

Using the matrix (inverse) method, solve the simultaneous equations 2x + 3y = 12 and x − y = 1.

  1. Write in matrix form: [ 2 3 ; 1 -1 ][ x ; y ] = [ 12 ; 1 ].
  2. Determinant of the coefficient matrix = (2×(−1)) − (3×1) = −2 − 3 = −5.
  3. Inverse = (1÷(−5))[ -1 -3 ; -1 2 ].
  4. So [ x ; y ] = (1÷(−5))[ -1 -3 ; -1 2 ][ 12 ; 1 ].
  5. Compute the product column: (−1×12)+(−3×1) = −15 and (−1×12)+(2×1) = −10.
  6. Then [ x ; y ] = (1÷(−5))[ -15 ; -10 ] = [ 3 ; 2 ].

Question 4 (6 marks)

Given A = [ 3 -1 ; 2 1 ]. (a) Find the determinant of A.

(b) Find A⁻¹. (c) Hence, using A⁻¹, solve 3x − y = 5 and 2x + y = 10.

  1. (a) Determinant = ad − bc = (3×1) − ((−1)×2) = 3 + 2 = 5.
  2. (b) Inverse = (1÷5)[ d -b ; -c a ] = (1÷5)[ 1 1 ; -2 3 ].
  3. (c) The equations give A[ x ; y ] = [ 5 ; 10 ], so [ x ; y ] = A⁻¹[ 5 ; 10 ].
  4. [ x ; y ] = (1÷5)[ 1 1 ; -2 3 ][ 5 ; 10 ].
  5. Product column: (1×5)+(1×10) = 15 and (−2×5)+(3×10) = 20.
  6. So [ x ; y ] = (1÷5)[ 15 ; 20 ] = [ 3 ; 4 ].

Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)

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Frequently asked questions

How are the Paper 1 and Paper 2 questions on matrices structured?

Paper 1 questions are multiple-choice, checking quick matrix addition, subtraction, or multiplication, and identifying whether a given matrix is a valid inverse. Paper 2 questions are structured, usually asking you to find a matrix inverse and use it to solve a pair of simultaneous linear equations.

Why should I attempt matrix questions before checking the worked solution?

Matrix operations follow a strict order, rows by columns, correct sign in the determinant, and attempting the question yourself is the only way to find out where your own working goes wrong. Seeing the final answer without struggling through the steps will not fix a habit of misordering multiplication.

What mistakes do students often make with matrix questions?

A frequent error is multiplying matrices in the wrong order, since matrix multiplication is not commutative, or writing the inverse formula with the determinant in the wrong position. Students also sometimes swap the wrong pair of elements when forming the adjoint of a 2 by 2 matrix.

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