Matrices and Determinants

Maths, HSSC-Punjab First Year. This chapter is split into 56 topics and has 1,328 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Scalar multiplication of a matrix 21 questions
2. Solving systems by Cramer's rule 27 questions
3. Principal and secondary diagonals of a square matrix 21 questions
4. Solving a linear system by the matrix inversion method 22 questions
5. Null matrices and equality of matrices 21 questions
6. Elementary row operations on a matrix 30 questions
7. Order of a matrix and position of elements 24 questions
8. Reflection (transformation) matrices 22 questions
9. Determinant of order 2 and expansion of order 3 along the first row 22 questions
10. Cramer's rule: statement and numerators 25 questions
11. Solving non-homogeneous systems by reduced echelon form 24 questions
12. Evaluating determinants of order 3 25 questions
13. Echelon form of a matrix 27 questions
14. Singular and non-singular matrices 21 questions
15. Solving equations given as determinants 21 questions
16. Linear systems: consistent, inconsistent and methods of solution 23 questions
17. Conformability and order of a matrix product 24 questions
18. Expansion of a determinant along any row or column 22 questions
19. Basic properties of determinants 27 questions
20. Matrices with complex entries 25 questions
21. Expansions of (A+B) squared and (A+B)(A-B) for matrices 22 questions
22. Reflecting points and triangles with transformation matrices 23 questions
23. Transpose of a matrix 24 questions
24. Addition and subtraction of matrices 26 questions
25. Rank of a matrix: definition and properties 24 questions
26. Rank and the nature of solutions of a linear system 23 questions
27. Upper, lower and triangular matrices 26 questions
28. Properties of matrix multiplication 24 questions
29. Row, column, rectangular and square matrices 25 questions
30. Product of 3x3 matrices 20 questions
31. Matrix multiplication is not commutative 22 questions
32. Powers of a square matrix 22 questions
33. Determinant of scalar multiples, transposes and products 22 questions
34. Homogeneous systems: trivial and non-trivial solutions 21 questions
35. Showing a determinant is zero without expansion 24 questions
36. Transpose of sums and products 23 questions
37. Minors of elements of a matrix 23 questions
38. Diagonal, scalar and identity matrices 22 questions
39. Determinants with algebraic entries and standard factorisations 23 questions
40. Product of rectangular matrices 20 questions
41. Inverse of a square matrix using the adjoint 26 questions
42. Coding and decoding messages using matrices 29 questions
43. Reduced row echelon form 22 questions
44. Cofactors of elements of a matrix 22 questions
45. Product of 2x2 matrices 25 questions
46. Cofactor matrix and adjoint of a square matrix 26 questions
47. Properties of the inverse and verifying A inverse times A 24 questions
48. Inverse of a matrix by elementary row operations 26 questions
49. Scaling matrices and image of a curve under a transformation 26 questions
50. Properties of matrix addition and scalar multiplication 25 questions
51. Row operations, splitting and triangular determinants 26 questions
52. Gaussian elimination for homogeneous systems 27 questions
53. Products with the identity and null matrices 23 questions
54. Matrix equations and matrix polynomials 23 questions
55. History and uses of matrices and determinants 25 questions
56. Computing the rank of a matrix 20 questions

Sample questions: Matrices and Determinants

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. For A = [4 −3 2; 2 1 4; 5 −4 −3], the sum a₁₁A₂₁ + a₁₂A₂₂ + a₁₃A₂₃ (row 1 entries with the cofactors of row 2) equals (rows separated by semicolons)

• A. 1
• B. 52
• C. 0(correct answer)
• D. −52

Answer: C.Entries of one row combined with the cofactors of a different row give 0, since this equals the determinant of a matrix with two identical rows (standard result). Here the sum is 0.

Q2. For A = [2 −1 −2; 1 1 −1; −3 −2 −4], the sum a₃₁A₂₁ + a₃₂A₂₂ + a₃₃A₂₃ (row 3 entries with the cofactors of row 2) equals (rows separated by semicolons)

• A. 1
• B. −21
• C. 0(correct answer)
• D. 21

Answer: C.Entries of one row combined with the cofactors of a different row give 0, since this equals the determinant of a matrix with two identical rows (standard result). Here the sum is 0.

Q3. For A = [4 2 4; 4 5 2; 3 −4 5], the sum a₂₁A₁₁ + a₂₂A₁₂ + a₂₃A₁₃ (row 2 entries with the cofactors of row 1) equals (rows separated by semicolons)

• A. 20
• B. 0(correct answer)
• C. 1
• D. −20

Answer: B.Entries of one row combined with the cofactors of a different row give 0, since this equals the determinant of a matrix with two identical rows (standard result). Here the sum is 0.

Q4. For A = [1 0 −1; 4 3 2; −1 2 4], the sum a₁₁A₂₁ + a₁₂A₂₂ + a₁₃A₂₃ (row 1 entries with the cofactors of row 2) equals (rows separated by semicolons)

• A. 0(correct answer)
• B. −3
• C. 3
• D. 1

Answer: A.Entries of one row combined with the cofactors of a different row give 0, since this equals the determinant of a matrix with two identical rows (standard result). Here the sum is 0.

Q5. For A = [5 3 −2; 4 −1 3; 5 −3 5], the sum a₂₁A₃₁ + a₂₂A₃₂ + a₂₃A₃₃ (row 2 entries with the cofactors of row 3) equals (rows separated by semicolons)

• A. 19
• B. 0(correct answer)
• C. −19
• D. 1

Answer: B.Entries of one row combined with the cofactors of a different row give 0, since this equals the determinant of a matrix with two identical rows (standard result). Here the sum is 0.

Q6. For A = [0 2 −3; 4 −4 3; 5 0 4], the sum a₂₁A₁₁ + a₂₂A₁₂ + a₂₃A₁₃ (row 2 entries with the cofactors of row 1) equals (rows separated by semicolons)

• A. 1
• B. 62
• C. 0(correct answer)
• D. −62

Answer: C.Entries of one row combined with the cofactors of a different row give 0, since this equals the determinant of a matrix with two identical rows (standard result). Here the sum is 0.

Q7. (Exercise 4.2 Q1(iii)) The determinant of [2x x x; y 2y y; z z 2z] equals (rows separated by semicolons)

• A. 2xyz
• B. 8xyz
• C. 6xyz
• D. 4xyz(correct answer)

Answer: D.Taking the common factors x, y and z out of rows 1, 2 and 3 gives xyz times the determinant of [2 1 1; 1 2 1; 1 1 2], which is 4, so the value is 4xyz (the book's key).

Q8. Using row operations on [A : I], the inverse of A = [−1 2 1; 2 −1 1; −2 −2 −2] is (rows separated by semicolons)

• A. [4 2 3; 2 4 3; −6 −6 −3]
• B. [−2/3 −1/3 −1/2; −1/3 −2/3 −1/2; 1 1 1/2](correct answer)
• C. [1/6 −1/3 1/3; −1/3 1/6 1/3; −1/6 −1/6 1/3]
• D. [−2/3 −1/3 −1/2; −1/3 −4/3 −1/2; 1 1 1/2]

Answer: B.Row-reduce [A : I] until the left block is I; the right block is A⁻¹ = [−2/3 −1/3 −1/2; −1/3 −2/3 −1/2; 1 1 1/2], in agreement with adj A / |A| for |A| = −6.

Q9. Using row operations on [A : I], the inverse of A = [1 0 2; 2 2 0; 1 −1 1] is (rows separated by semicolons)

• A. [−1/6 −1/3 −1/6; 0 −1/3 1/6; −1/3 0 −1/6]
• B. [−1/3 1/3 2/3; 1/3 1/6 −2/3; 2/3 −1/6 −1/3](correct answer)
• C. [−1/3 1/3 2/3; 1/3 1/6 −2/3; −2/3 −1/6 −1/3]
• D. [−1/3 1/3 2/3; 1/3 −11/6 −2/3; 2/3 −1/6 −1/3]

Answer: B.Row-reduce [A : I] until the left block is I; the right block is A⁻¹ = [−1/3 1/3 2/3; 1/3 1/6 −2/3; 2/3 −1/6 −1/3], in agreement with adj A / |A| for |A| = −6.

Q10. For 3 unknowns, rank A = 2 and rank [A : B] = 2. Such a system has

• A. exactly two solutions
• B. exactly one (unique) solution
• C. no solution (inconsistent)
• D. infinitely many solutions(correct answer)

Answer: D.Compare the ranks: rank A = 2, rank [A : B] = 2, variables = 3; they are equal but less than the number of variables, so there are infinitely many solutions.