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Matrices and Systems of Linear Equations · CAPE Pure Mathematics Unit 2

112 past-paper questions on Matrices and Systems of Linear Equations, part of Counting, Matrices and Differential Equations, from every CAPE Pure Mathematics Unit 2 paper on Quelpr.

Practise Matrices and Systems of Linear Equations
  1. 6(a)10 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Find the values of x for which the determinant of the 3x3 matrix with rows [x, 1, 2], [1, x, 2], and [2, 1, x] is equal to 0.
  2. 6(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Find AB.
  3. 6(a)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Deduce A^(-1).
  4. 6(b)(i)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Write down an expression in terms of p, q and r, for the number of kilograms of Z-grass in the blend.
  5. 6(b)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Let c, z and b represent the number of kilograms of C-grass, Z-grass and B-grass respectively in the blend. Write down a set of THREE equations in p, q, r, to represent the number of kilograms of EACH type of grass in…
  6. 6(b)(iii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Rewrite the set of THREE equations in (b)(ii) above in the matrix form MX = D where M is a 3 by 3 matrix, X and D are column matrices.
  7. 6(b)(iv)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Given that M^(-1) exists, write X in terms of M^(-1) and D.
  8. 6(b)(v)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Given that M^(-1) = [[-0.2, -0.2, 0.3], [0.35, 0.1, -0.15], [-0.05, 0.2, -0.05]], calculate how many bags of EACH brand, P, Q, and R, are required to produce a blend containing 30 kilograms of C-grass, 30 kilograms of…
  9. 6(a)(i)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Write the system in matrix form.
  10. 6(a)(ii)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Write down the augmented matrix.
  11. 6(a)(iii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Reduce the augmented matrix to echelon form.
  12. 6(a)(iv)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Deduce the value of \alpha for which the system is consistent.
  13. 6(a)(v)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Find ALL solutions corresponding to this value of \alpha.
  14. 6(b)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Find kI - A, where I is the 3 \times 3 identity matrix and k is a real number.
  15. 6(b)(ii)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Find the values of k for which |kI - A| = 0.
  16. 5(b)(i)a)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Determine the matrix A - B.
  17. 5(b)(i)b)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Determine the matrix AM.
  18. 5(b)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Deduce from (i) b) above the inverse A^{-1} of the matrix A.
  19. 5(b)(iii)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Find the matrix X such that AX + B = A.
  20. 6(a)12 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Solve for x the equation \begin{vmatrix} 1 & 1 & 1 \\ x & 2 & 1 \\ x^3 & 8 & 1 \end{vmatrix} = 0.
  21. 6(b)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Express the information above as a matrix equation AX = Y, where A is a 3 \times 3 matrix, and X and Y are 3 \times 1 matrices with X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}.
  22. 6(b)(ii)a)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Calculate AB.
  23. 6(b)(ii)b)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Deduce the inverse A^{-1} of A.
  24. 6(b)(iii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Hence, or otherwise, determine the number of cars and buses used in the 34\text{ km} tours.
  25. 6(a)10 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Solve for x the equation \begin{vmatrix} x - 3 & 1 & -1 \\ 1 & x - 5 & 1 \\ -1 & 1 & x - 3 \end{vmatrix} = 0.
  26. 6(b)(i)a)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Find \mathbf{A}\mathbf{B}.
  27. 6(b)(i)b)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Hence deduce the inverse \mathbf{A}^{-1} of the matrix \mathbf{A}.
  28. 6(b)(ii)a)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Express the system in the form \mathbf{A}\mathbf{x} = \mathbf{b}, where \mathbf{A} is a matrix and \mathbf{x} and \mathbf{b} are column vectors.
  29. 6(b)(ii)b)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Hence, or otherwise, solve the system of equations.
  30. 6(a)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Write the augmented matrix of the system.
  31. 6(a)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Reduce the augmented matrix to echelon form.
  32. 6(a)(iii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Deduce the value of k for which the system is consistent.
  33. 6(a)(iv)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Find ALL solutions corresponding to the value of k obtained in (iii) above.
  34. 6(b)(i)a)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Find A^2.
  35. 6(b)(i)b)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Find B = 3I + A - A^2.
  36. 6(b)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Calculate AB.
  37. 6(b)(iii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Deduce the inverse, A^{-1}, of the matrix A.
  38. 6(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Show that |A| = 5.
  39. 6(a)(ii) a)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Matrix B is formed by interchanging row 1 and row 2 of matrix A and then interchanging column 1 and column 2 of the resulting matrix. Write down det(B), giving a reason.
  40. 6(a)(ii) b)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Row 1 of matrix C is formed by adding row 2 to row 1 of matrix A. The other rows remain unchanged. Write down det(C), giving a reason.
  41. 6(a)(ii) c)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Matrix D is formed by multiplying each element of matrix A by 5. Write down det(D), giving a reason.
  42. 6(b)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Find AM.
  43. 6(b)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Find the inverse, A^{-1}, of A.
  44. 6(c)(i)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Write the system of equations in the form Ax = b.
  45. 6(c)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Show that x = A^{-1}b.
  46. 6(c)(iii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Hence, solve the system of equations.
  47. 6(c)(iv) a)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Show that (x, y, z) = (1, 1, 1) is a solution of the system of equations: x + y + z = 3 2x + 2y + 2z = 6 3x + 3y + 3z = 9.
  48. 6(c)(iv) b)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Hence, find the general solution of the system.
  49. 5(c)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Find the matrix \mathbf{B}, where \mathbf{B} = \mathbf{A}^2 - 3\mathbf{A} - \mathbf{I}.
  50. 5(c)(ii)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Show that \mathbf{AB} = -9\mathbf{I}.
  51. 5(c)(iii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Hence, find the inverse, \mathbf{A}^{-1}, of \mathbf{A}.
  52. 5(c)(iv)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Solve the system of linear equations \mathbf{B} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}.
  53. 5(b)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2The augmented matrix for a system of three linear equations with variables x, y and z respectively is A = \begin{pmatrix} 1 & 1 & -1 & | & 1 \\ -5 & 1 & 1 & | & 2 \\ 1 & -5 & 3 & | & 3 \end{pmatrix}. By reducing…
  54. 5(b)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2The augmented matrix for another system is formed by replacing the THIRD row of A in (i) above with (1 \; -5 \; 5 \mid 3). Determine whether the solution of the new system is unique. Give a reason for your answer.
  55. 5(b)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Determine the range of values of x for which A^{-1} exists.
  56. 5(b)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Given that \det(AB) = -21, show that x = 3.
  57. 5(b)(iii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Hence, obtain A^{-1}.
  58. Q351 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The matrix A is a 3 \times 3 matrix with determinant 14. If the matrix of cofactors of A is \begin{pmatrix} 4 & -14 & -2 \\ 3 & -7 & -5 \\ 1 & 7 & 3 \end{pmatrix}, then A^{-1} =
  59. Q361 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The number of possible values of x which satisfy the system of simultaneous equations, \begin{align*} 2x + 3y + 2z &= -5 \\ 4x + 6y + 4z &= -10 \\ 6x + 9y + 6z &= -16 \end{align*} is
  60. Q371 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1If M = \begin{pmatrix} 1 & 1 & 4 \\ 3 & 2 & -1 \\ 6 & 0 & 5 \end{pmatrix}, then the cofactor of the element 3 in M above may be written as
  61. Q381 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1If P = \begin{pmatrix} 1 & -2 & 0 \\ 3 & 1 & 5 \\ -1 & 2 & 3 \end{pmatrix} and Q = \begin{pmatrix} -7 & 6 & -10 \\ -14 & 3 & -5 \\ 7 & 0 & 7 \end{pmatrix} then…
  62. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The determinant of the matrix M = \begin{pmatrix} 3 & -1 & 5 \\ 2 & 3 & -2 \\ 0 & 5 & 4 \end{pmatrix} is
  63. 5(c)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Rewrite the system of equations as an augmented matrix.
  64. 5(c)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Use elementary row operations to reduce the system to echelon form.
  65. 5(c)(iii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Hence, solve the system of equations.
  66. 5(c)(iv)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Show that the system has no solution if the third equation is changed to 1.5x - 1.5y + 3z = 9.
  67. Q321 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1A is a 3 \times 3 matrix with determinant 14. If the matrix of cofactors of A is \begin{pmatrix} 4 & -14 & -2 \\ 3 & -7 & -5 \\ 1 & 7 & 3 \end{pmatrix}, then A^{-1} =
  68. Q331 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1If P = \begin{pmatrix} 1 & -2 & 0 \\ 3 & 1 & 5 \\ -1 & 2 & 3 \end{pmatrix} and Q = \begin{pmatrix} -7 & 6 & -10 \\ -14 & 3 & -5 \\ 7 & 0 & 7 \end{pmatrix}, then…
  69. Q381 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1The FIRST ROW of the product PQ of the two 3 \times 3 matrices P = \begin{pmatrix} 2 & 3 & 1 \\ 5 & -6 & 5 \\ -1 & 2 & 3 \end{pmatrix} and Q = \begin{pmatrix} 2 & 1 & 3 \\ 5 & 0 & -1 \\ -3 & -2 & 4 \end{pmatrix}…
  70. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1A, B, C and D are four 3 \times 3 matrices. Given that AB = J, BC = K, CD = L, ABC = P and BCD = Q, where J, K, L, P and Q are matrices, the product of ABCD is
  71. Q441 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1The number of possible values of x which satisfy the system of simultaneous equations, \[\begin{aligned} 2x + 3y + 2z &= -5 \\ 4x + 6y + 4z &= -10 \\ 6x + 9y + 6z &= -25 \end{aligned}\] is
  72. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1The matrix A represents a system of linear equations after some elementary row operations have been performed. \[A = \begin{pmatrix} 1 & 0 & 1 & : & 3 \\ 0 & 1 & -1 & : & 2 \\ 0 & 0 & 0 & : & 2 \end{pmatrix}\] Which…
  73. 5(c)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Find |A|, the determinant of A.
  74. 5(c)(ii)10 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Hence, or otherwise, find A^(-1), the inverse of A.
  75. Q321 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1If P = \begin{pmatrix} 1 & -2 & 0 \\ 3 & 1 & 5 \\ -1 & 2 & 3 \end{pmatrix} and Q = \begin{pmatrix} -7 & 6 & -10 \\ -14 & 3 & -5 \\ 7 & 0 & 7 \end{pmatrix}, then…
  76. Q341 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The determinant of the matrix M = \begin{pmatrix} 3 & -1 & 5 \\ 2 & 3 & -2 \\ 0 & 5 & 4 \end{pmatrix} is
  77. Q361 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Given that H is a non-singular, square matrix, the determinant |H^2| of H^2 is
  78. Q371 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The number of possible values of x which satisfy the system of simultaneous equations, \begin{aligned} 2x + 3y + 2z &= -5 \\ 4x + 6y + 4z &= -10 \\ 6x + 9y + 6z &= -16 \end{aligned} is
  79. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The planes represented by the equations \begin{aligned} 2x + y - z &= 4 \\ x + y + z &= 1 \\ 3x - 2y - z &= 2 \end{aligned} I. are inconsistent II. are not parallel III. have a unique solution
  80. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The matrix \mathbf{A} represents a system of linear equations after some elementary row operations have been performed. \mathbf{A} = \begin{pmatrix} 1 & 0 & 1 & : & 3 \\ 0 & 1 & -1 & : & 2 \\ 0 & 0 & 0 & : & 2…
  81. 5(c)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Show that AB = 20I.
  82. 5(c)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Hence, deduce the inverse, A^{-1}, of the matrix A.
  83. 5(c)(iii)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Hence, or otherwise, solve the system of linear equations given by: \begin{aligned} x - y + z &= 1 \\ x - 2y + 4z &= 5 \\ x + 3y + 9z &= 25 \end{aligned}
  84. Q311 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1If \mathbf{M} = \begin{pmatrix} 1 & 3 & 3 \\ -2 & 4 & 1 \\ 0 & 5 & 6 \end{pmatrix}, the FIRST ROW of the co-factor matrix of \mathbf{M} is
  85. Q321 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The determinant of the matrix \mathbf{M} = \begin{pmatrix} 3 & -1 & 5 \\ 2 & 3 & -2 \\ 0 & 5 & 4 \end{pmatrix} is
  86. Q371 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1If \mathbf{M} = \begin{pmatrix} 1 & 1 & 4 \\ 3 & 2 & -1 \\ 6 & 0 & 5 \end{pmatrix}, then the co-factor of the element 3 in \mathbf{M} above may be written as
  87. Q381 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The number of possible values of x which satisfy the system of simultaneous equations \begin{aligned} 2x + 3y + 2z &= -5 \\ 4x + 6y + 4z &= -10 \\ 6x + 9y + 6z &= -25 \end{aligned} is
  88. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1\mathbf{A}, \mathbf{B}, \mathbf{C} and \mathbf{D} are four 3 \times 3 matrices. Given that \mathbf{AB} = \mathbf{J}, \mathbf{BC} = \mathbf{K}, \mathbf{CD} = \mathbf{L}, \mathbf{ABC} = \mathbf{P} and…
  89. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The matrix \mathbf{A} represents a system of linear equations after some elementary row operations have been performed. \mathbf{A} = \begin{pmatrix} 1 & 0 & 1 & : & 3 \\ 0 & 1 & -1 & : & 2 \\ 0 & 0 & 0 & : & 2…
  90. 5(d)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2By finding AB, deduce that A^(-1) = 1/88 B.
  91. 5(d)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Hence, or otherwise, solve the system of equations given by [[5, -2, 3], [0, 3, -4], [2, 0, 6]] [[x], [y], [z]] = [[7], [11], [-6]].
  92. Q361 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1Given that \mathbf{H} is a non-singular, square matrix, the determinant |\mathbf{H}^2| of \mathbf{H}^2 is
  93. Q391 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The planes represented by the equations \begin{aligned} 2x + y - z &= 4 \\ x + y + z &= 1 \\ 3x - 2y - z &= 2 \end{aligned} \nI. are inconsistent\nII. are not parallel\nIII. have a unique solution
  94. Q401 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The number of possible values of x which satisfy the system of simultaneous equations \begin{aligned} 2x + 3y + 2z &= -5 \\ 4x + 6y + 4z &= -10 \\ 6x + 9y + 6z &= -16 \end{aligned}\nis
  95. Q411 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The matrix \mathbf{M} = \begin{pmatrix} 2 & -5 & 6 \\ 1 & 1 & -2 \\ a & 2 & 2 \end{pmatrix}. \nIf \det \mathbf{M} = 22, then a =
  96. Q451 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1Item 45 refers to the matrix \mathbf{M} below. \mathbf{M} = \begin{pmatrix} 1 & 2 & 4 \\ -1 & 3 & 0 \\ 0 & 1 & 5 \end{pmatrix} \nThe co-factor of the element 3 in \mathbf{M} may be written as
  97. 5(a)6 marks· Pure Mathematics · Unit 2 Q5 5(a)Given that the matrix A = [[1-p, 3, -2], [2, p, -1], [-3, 2, 1]] is singular, calculate the value(s) of p.
  98. 5(a)(i)5 marks· Pure Mathematics · Unit 2 Q5 5(a)(i)Calculate the determinant of M.
  99. 5(a)(i)4 marks· Pure Mathematics · Unit 2 Q5 5(a)(i)Show that |P| = 183.
  100. 5(a)(ii)5 marks· Pure Mathematics · Unit 2 Q5 5(a)(ii)Hence, or otherwise, show that the adjoint of P is adj(P) = [[-162, 59, 25], [207, -72, -15], [42, -4, -11]].
  101. 5(a)(ii)4 marks· Pure Mathematics · Unit 2 Q5 5(a)(ii)Hence, or otherwise, determine the values of k for which the simultaneous equations x + y - z = 1, x + 2y - kz = 0, x - ky - z = 1 have a unique solution.
  102. 5(a)(iii)8 marks· Pure Mathematics · Unit 2 Q5 5(a)(iii)Solve the system of linear equations [[4, 3, 5], [9, 4, 15], [12, 10, -3]] [[x], [y], [z]] = [[11], [13], [4]].
  103. 5(a)(iii)6 marks· Pure Mathematics · Unit 2 Q5 5(a)(iii)Using k = 2 in the matrix, M, solve the system of linear equations by first reducing it to row echelon form.
  104. 5(b)9 marks· Pure Mathematics · Unit 2 Q5 5(b)Use row reduction to solve the following system of equations: 3x - 4y + z = 16; 2x + y - 2z = 5; x + 2y - z = -2.
  105. 5(b)(i)4 marks· Pure Mathematics · Unit 2 Q5 5(b)(i)Determine the value of x for which A⁻¹ does NOT exist.
  106. 5(b)(ii)4 marks· Pure Mathematics · Unit 2 Q5 5(b)(ii)Given that det(AB) = -10, show that x = 2.
  107. 5(b)(iii)4 marks· Pure Mathematics · Unit 2 Q5 5(b)(iii)Hence, obtain A⁻¹.
  108. 5(c)(i)4 marks· Pure Mathematics · Unit 2 Q5 5(c)(i)By reducing the matrix to row echelon form, show that the system has a finite set of solutions.
  109. 5(c)(ii)4 marks· Pure Mathematics · Unit 2 Q5 5(c)(ii)Hence, solve the system of linear equations.
  110. 5(d)4 marks· Pure Mathematics · Unit 2 Q5 5(d)Evaluate the determinant of A.
  111. 5(e)5 marks· Pure Mathematics · Unit 2 Q5 5(e)Determine whether the system is consistent.
  112. 6(a)4 marks· Pure Mathematics · Unit 2 Q6 6(a)Calculate the value of k for which the matrix is singular.

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