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Matrices and determinants: board exam questions with solutions

On the ECE exam under Mathematics and the REE exam under Mathematics. Try each question before you open its solution, then practise with fresh numbers below: 1 computed question type on this topic.

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Questions

1. Find the determinant of the 2×2 matrix with rows [3 2] and [1 4].

  1. 14
  2. 10
  3. −10
  4. 5
Show the answer and solution

Answer: B. 10

det = 3(4) − 2(1) = 12 − 2 = 10.

2. A is 2×3 and B is 3×4. What is the size of AB?

  1. 3×3
  2. 4×2
  3. The product is undefined
  4. 2×4
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Answer: D. 2×4

The inner dimensions (3) match, and the product takes the outer dimensions: 2×4.

3. Find the determinant of the matrix with rows [1 2 3], [0 4 5] and [1 0 6].

  1. 24
  2. −22
  3. 22
  4. 10
Show the answer and solution

Answer: C. 22

Expanding along row 1: 1(24 − 0) − 2(0 − 5) + 3(0 − 4) = 24 + 10 − 12 = 22.

4. Find the inverse of the matrix with rows [2 1] and [5 3].

  1. Rows [3 −5] and [−1 2]
  2. Rows [−3 1] and [5 −2]
  3. Rows [3 −1] and [−5 2]
  4. Rows [2 −1] and [−5 3]
Show the answer and solution

Answer: C. Rows [3 −1] and [−5 2]

det = 6 − 5 = 1. The inverse is (1/det)[d −b; −c a] = [3 −1; −5 2].

5. Solve for y: 2x + y = 7 and x − y = −1.

  1. 3
  2. 2
  3. 5
  4. −3
Show the answer and solution

Answer: A. 3

Adding the equations gives 3x = 6, so x = 2 and y = 7 − 2(2) = 3. By Cramer's rule: D = −3, D_y = −9, y = 3.

6. Find the eigenvalues of the matrix with rows [4 1] and [2 3].

  1. 4 and 3
  2. 5 and 2
  3. 7 and 10
  4. 6 and 1
Show the answer and solution

Answer: B. 5 and 2

det(A − λI) = (4 − λ)(3 − λ) − 2 = λ² − 7λ + 10 = 0, so λ = 5 or λ = 2.

7. Mesh analysis of a circuit gives 10I₁ − 4I₂ = 12 and −4I₁ + 8I₂ = 0, with currents in amperes. Find I₁.

  1. 0.75 A
  2. 1.2 A
  3. 3.0 A
  4. 1.5 A
Show the answer and solution

Answer: D. 1.5 A

The second equation gives I₁ = 2I₂. Substituting: 20I₂ − 4I₂ = 12, so I₂ = 0.75 A and I₁ = 1.5 A. 1.2 A comes from ignoring the coupling term.

8. For conformable matrices A and B, (AB)ᵀ equals:

  1. BᵀAᵀ
  2. AᵀBᵀ
  3. BA
  4. (BA)ᵀ
Show the answer and solution

Answer: A. BᵀAᵀ

Taking the transpose of a product reverses the order: (AB)ᵀ = BᵀAᵀ.

Worked examples with computed numbers

These numbers are fixed on this page; in practice they change every time.

9. Find the determinant of the matrix [0 3 6; 2 1 1; 4 -1 -5].

  1. -24
  2. 6
  3. -6
  4. -78
Show the answer and solution

Answer: B. 6

Expand along the first row: 0(-4) − 3(-14) + 6(-6) = 6.

Other topics in the same subjects

Algebra · Trigonometry · Analytic geometry · Differential calculus · Integral calculus · Differential equations · Complex numbers · Probability and statistics · Laplace transforms and Fourier series

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