EXM.2.AHL.TZ0.10

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

Let A = $\left( {\begin{array}{*{20}{c}} 0&2 \\ 2&0 \end{array}} \right)$ .

Let B = $\left( {\begin{array}{*{20}{c}} p&2 \\ 0&q \end{array}} \right)$ .

Find A⁻¹.

[2]

a.i.

Find A².

[2]

a.ii.

Given that 2A + B = $\left( {\begin{array}{*{20}{c}} 2&6 \\ 4&3 \end{array}} \right)$ , find the value of $p$ and of $q$ .

[3]

Hence find A⁻¹B.

[2]

Let X be a 2 × 2 matrix such that AX = B. 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 = $\left( {\begin{array}{*{20}{c}} 0&{\frac{1}{2}} \\ {\frac{1}{2}}&0 \end{array}} \right)$ A2 N2

[2 marks]

a.i.

A² = $\left( {\begin{array}{*{20}{c}} 4&0 \\ 0&4 \end{array}} \right)$ A2 N2

[2 marks]

a.ii.

$\left( {\begin{array}{*{20}{c}} 0&4 \\ 4&0 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} p&2 \\ 0&q \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 2&6 \\ 4&3 \end{array}} \right)$ (M1)

$p$ = 2, $q$ = 3 A1A1 N3

Evidence of attempt to multiply (M1)

eg A⁻¹B = $\left( {\begin{array}{*{20}{c}} 0&{\frac{1}{2}} \\ {\frac{1}{2}}&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&2 \\ 0&3 \end{array}} \right)$

A⁻¹B = $\left( {\begin{array}{*{20}{c}} 0&{\frac{3}{2}} \\ 1&1 \end{array}} \right)$ $\left( {{\text{accept}}\left( {\begin{array}{*{20}{c}} 0&{\frac{1}{2}q} \\ {\frac{1}{2}p}&1 \end{array}} \right)} \right)$ A1 N2

[2 marks]

Evidence of correct approach (M1)

eg X = A⁻¹B, setting up a system of equations

X = $\left( {\begin{array}{*{20}{c}} 0&{\frac{3}{2}} \\ 1&1 \end{array}} \right)$ $\left( {{\text{accept}}\left( {\begin{array}{*{20}{c}} 0&{\frac{1}{2}q} \\ {\frac{1}{2}p}&1 \end{array}} \right)} \right)$ A1 N2

[2 marks]

Examiners’ report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]