EXM.2.AHL.TZ0.6

pestleMathematicsAIHLPaper 2EXM· ahl-1-14-introduction-to-matricessource ↗

Matrices A, B and C are defined by

A = $\left( {\begin{array}{*{20}{c}} 5&1 \\ 7&2 \end{array}} \right)$ B = $\left( {\begin{array}{*{20}{c}} 2&4 \\ { - 3}&{15} \end{array}} \right)$ C = $\left( {\begin{array}{*{20}{c}} 9&{ - 7} \\ 8&2 \end{array}} \right)$ .

Let X be an unknown 2 × 2 matrix satisfying the equation

AX + B = C.

This equation may be solved for X by rewriting it in the form

X = A⁻¹ D.

where D is a 2 × 2 matrix.

Write down A⁻¹.

[2]

Find D.

[3]

Find X.

[2]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

A⁻¹ = $\left( {\begin{array}{*{20}{c}} {\frac{2}{3}}&{ - \frac{1}{3}} \\ { - \frac{7}{3}}&{\frac{5}{3}} \end{array}} \right)$ or $\frac{1}{3}\left( {\begin{array}{*{20}{c}} 2&{ - 1} \\ { - 7}&5 \end{array}} \right)$ or $\left( {\begin{array}{*{20}{c}} {0.667}&{ - 0.333} \\ { - 2.33}&{ - 1.67} \end{array}} \right)$ (A1)(A1)(N2)

[2 marks]

AX = C − B (may be implied) (A1)

X = A⁻¹(C−B) (A1)

D = C − B

$= \left( {\begin{array}{*{20}{c}} 7&{ - 11} \\ {11}&{ - 13} \end{array}} \right)$ (A1) (N3)

[3 marks]

X = $\left( {\begin{array}{*{20}{c}} 1&{ - 3} \\ 2&4 \end{array}} \right)$ (A2) (N2)

[2 marks]

Examiners’ report

[N/A]