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2 marksMatrices and Systems of Linear Equations

CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2 · Question 6(b)(ii)

Given the matrix M = \begin{pmatrix} 12 & -1 & 5 \\ 2 & -1 & 0 \\ -9 & 2 & -5 \end{pmatrix}.

Find the inverse, A^{-1}, of A.

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  1. 6(a)(i)Show that |A| = 5.[3 marks]
  2. 6(a)(ii) a)Matrix 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…[2 marks]
  3. 6(a)(ii) b)Row 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.[2 marks]
  4. 6(a)(ii) c)Matrix D is formed by multiplying each element of matrix A by 5. Write down det(D), giving a reason.[2 marks]
  5. 6(b)(i)Find AM.[3 marks]
  6. 6(c)(i)Write the system of equations in the form Ax = b.[1 mark]
  7. 6(c)(ii)Show that x = A^{-1}b.[2 marks]
  8. 6(c)(iii)Hence, solve the system of equations.[2 marks]
  9. 6(c)(iv) a)Show 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.[1 mark]
  10. 6(c)(iv) b)Hence, find the general solution of the system.[5 marks]

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