Matrices and Determinants

Maths, HSSC-KPK First Year. This chapter is split into 66 topics and has 1,349 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Product of 3 by 3 matrices 20 questions
2. Commutative property of matrix addition 20 questions
3. Elementary column operations 20 questions
4. Finding the inverse by elementary row operations 20 questions
5. Homogeneous and non-homogeneous linear equations 20 questions
6. Review Exercise 2: mixed problems 20 questions
7. Matrix form AX = B and the augmented matrix 20 questions
8. Property 5: multiplying a row by a constant 20 questions
9. Cramer's rule 20 questions
10. Echelon form of a matrix 20 questions
11. Singular and non-singular matrices 20 questions
12. Evaluating determinants using properties 20 questions
13. Transpose of a sum 20 questions
14. Properties of the inverse of a matrix 20 questions
15. Homogeneous systems: trivial and non-trivial solutions 20 questions
16. Matrix multiplication is not commutative 20 questions
17. Elementary row operations 22 questions
18. Adjoint of a square matrix 20 questions
19. Determinant of a 2 by 2 matrix 22 questions
20. Solving systems by the matrix inversion method 20 questions
21. Unit (identity) matrix and zero (null) matrix 20 questions
22. Properties of determinants: interchange and identical rows 20 questions
23. Reducing a matrix to echelon form 20 questions
24. Powers of a matrix and matrix polynomials 20 questions
25. Reduced echelon form by row reduction 20 questions
26. Subtraction of matrices 21 questions
27. Transpose of a matrix 21 questions
28. Diagonal and scalar matrices 20 questions
29. Determinant of a 3 by 3 matrix by expansion along a row 22 questions
30. Determinant of a 3 by 3 matrix by the diagonal (Sarrus) method 20 questions
31. Inverse of a 3 by 3 matrix by the adjoint method 20 questions
32. Products of row and column matrices 20 questions
33. Determinant equations in an unknown 20 questions
34. Upper and lower triangular matrices 21 questions
35. The products AAt and AtA 20 questions
36. Consistent and inconsistent systems 20 questions
37. Minors of a matrix 22 questions
38. Solving systems by Gauss elimination (echelon form) 20 questions
39. Product of rectangular matrices 22 questions
40. Conformability and order of a product 20 questions
41. Inverse of a 2 by 2 matrix 20 questions
42. Finding the parameter for a non-trivial solution 20 questions
43. Addition of matrices 22 questions
44. Reduced echelon form of a matrix 20 questions
45. Linear combinations of matrices 20 questions
46. Solving matrix equations for X 20 questions
47. Determinant by expansion along a column 21 questions
48. Equality of two matrices 20 questions
49. Associative and distributive laws for matrices 20 questions
50. Properties of determinants: zero row and transpose 20 questions
51. Cofactors of a matrix 22 questions
52. Properties 6 and 7: splitting a row and adding a multiple of a row 20 questions
53. Symmetric and skew-symmetric matrices 20 questions
54. Transpose of a product 20 questions
55. Solving systems by the Gauss-Jordan method 20 questions
56. Row matrices and column matrices 22 questions
57. Product of two 2 by 2 matrices 22 questions
58. Scalar multiplication of a matrix 21 questions
59. Concept of a matrix and its notation 22 questions
60. Square and rectangular matrices; the principal diagonal 22 questions
61. Order of a matrix and its elements 22 questions
62. A + At is symmetric and A minus At is skew-symmetric 20 questions
63. Rank of a matrix 20 questions
64. Applying elementary operations (Example 7) 20 questions
65. Finding unknowns in symmetric and skew-symmetric matrices 20 questions
66. Proving and checking determinant identities 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. A matrix is defined in the book as

• A. a square array whose rows and columns are equal in number
• B. a single row of numbers separated by commas
• C. a rectangular array of numbers enclosed in large square brackets or parentheses(correct answer)
• D. a number that is obtained from an array by multiplying its rows

Answer: C.The text defines a matrix as a rectangular array of numbers enclosed in large square brackets or parentheses; the book uses square brackets throughout.

Q2. The Did You Know box says that a matrix is merely

• A. a vector whose length is its order
• B. a table of numbers with no particular value in itself(correct answer)
• C. a number that represents the sum of its entries
• D. a number obtained by evaluating its determinant

Answer: B.The box says a matrix is merely a table of numbers: apart from being a convenient way of recording certain types of numerical values, it has no particular value in itself.

Q3. The introduction to Unit 2 says that matrices and matrix methods are widely applied in

• A. weather observation by hand only
• B. the writing of historical novels
• C. the operation of high speed computers(correct answer)
• D. the preparation of cooking recipes

Answer: C.Section 2.1 says matrix methods have widespread applications, for example in the operation of high speed computers, and in science, engineering and business.

Q4. The matrix [2 0 0; 8 8 0; 9 3 3] (rows separated by semicolons) is

• A. upper triangular
• B. lower triangular(correct answer)
• C. neither upper nor lower triangular
• D. both upper and lower triangular (diagonal)

Answer: B.Check the zeros: upper triangular needs zeros below the diagonal, lower triangular needs zeros above it; a diagonal matrix satisfies both.

Q5. The matrix [7 2 6 6; 0 7 2 2; 0 0 4 3; 3 0 0 4] (rows separated by semicolons) is

• A. lower triangular
• B. both upper and lower triangular (diagonal)
• C. neither upper nor lower triangular(correct answer)
• D. upper triangular

Answer: C.Check the zeros: upper triangular needs zeros below the diagonal, lower triangular needs zeros above it; a diagonal matrix satisfies both.

Q6. The matrix [9 0 0; 0 9 0; 0 0 3] (rows separated by semicolons) is

• A. neither upper nor lower triangular
• B. lower triangular
• C. both upper and lower triangular (diagonal)(correct answer)
• D. upper triangular

Answer: C.Check the zeros: upper triangular needs zeros below the diagonal, lower triangular needs zeros above it; a diagonal matrix satisfies both.

Q7. Which is true for A = [1 −3; 3 4] (rows separated by semicolons) and B = [5 1; 0 −1]?

• A. A + B = B + A(correct answer)
• B. A − B = B − A
• C. A + B = −(A + B)
• D. A + B = B − A

Answer: A.Matrix addition is commutative: both sums equal [6 −2; 3 3]; subtraction is not commutative (A − B = [−4 −4; 3 5], B − A = [4 4; −3 −5]).

Q8. Which operation changes the value of a determinant by a factor k?

• A. adding a multiple of one row to another
• B. multiplying one row by 4(correct answer)
• C. interchanging two rows once
• D. adding a multiple of one column to another

Answer: B.Only scaling a single row (or column) multiplies the determinant (by the scale factor); adding a multiple of a row leaves it unchanged, and interchanging only changes its sign.

Q9. Example 9: the inverse of A = [2 3 1; 5 4 2; −1 2 −2] (rows separated by semicolons) is

• A. (1/14)[−12 8 2; 8 −3 1; 14 −7 −7](correct answer)
• B. (1/14)[12 −8 −2; −8 3 −1; −14 7 7]
• C. [−12 8 14; 8 −3 −7; 2 1 −7]
• D. (1/14)[−11 8 2; 8 −3 1; 14 −7 −7]

Answer: A.Using A⁻¹ = adj A/|A| with |A| = 14: (1/14)[−12 8 2; 8 −3 1; 14 −7 −7]. (The final matrix printed in the book does not match this correct value, so the computed inverse is used here.)

Q10. The system −x − y = 2, 3x − 2y + 3z = −1, 5x − 5y + 6z = 0 is

• A. inconsistent
• B. consistent with a unique solution
• C. consistent with infinitely many solutions(correct answer)
• D. homogeneous with only the zero solution

Answer: C.The rank of the coefficient matrix is 2 and that of the augmented matrix is 2; with equal ranks below 3 there are infinitely many solutions.