EXM.1.AHL.TZ0.16

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

Find a relationship between $a$ and $b$ if the matrices $M = \left( {\begin{array}{*{20}{c}} 1&a \\ 2&3 \end{array}} \right)$ and $N = \left( {\begin{array}{*{20}{c}} 1&b \\ 2&3 \end{array}} \right)$ commute under matrix multiplication.

[4]

Find the value of $a$ if the determinant of matrix $M$ is −1.

[2]

b.i.

Write down ${M^{ - 1}}$ for this value of $a$ .

[1]

b.ii.

Markscheme / solution

$\left( {\begin{array}{*{20}{c}} 1&a \\ 2&3 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&b \\ 2&3 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {1 + 2a}&{b + 3a} \\ 8&{2b + 9} \end{array}} \right)$

$\left( {\begin{array}{*{20}{c}} 1&b \\ 2&3 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&a \\ 2&3 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {1 + 2b}&{a + 3b} \\ 8&{2a + 9} \end{array}} \right)$ M1A1

So require $a = b$ M1A1

[4 marks]

$\left| {\begin{array}{*{20}{c}} 1&a \\ 2&3 \end{array}} \right| = 3 - 2a = - 1 \Rightarrow a = 2$ M1A1

[2 marks]

b.i.

${\left( {\begin{array}{*{20}{c}} 1&2 \\ 2&3 \end{array}} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} { - 3}&2 \\ 2&{ - 1} \end{array}} \right)$ A1

[1 mark]

b.ii.

Examiners’ report

[N/A]

b.i.

[N/A]

b.ii.