EXM.1.AHL.TZ0.4

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

Consider the matrix A $= \left( {\begin{array}{*{20}{c}} 5&{ - 2} \\ 7&1 \end{array}} \right)$ .

B, C and X are also 2 × 2 matrices.

Write down the inverse, A^–1.

[2]

Given that XA + B = C, express X in terms of A^–1, B and C.

[2]

b.i.

Given that B $= \left( {\begin{array}{*{20}{c}} 6&7 \\ 5&{ - 2} \end{array}} \right)$ , and C $= \left( {\begin{array}{*{20}{c}} { - 5}&0 \\ { - 8}&7 \end{array}} \right)$ , find X.

[2]

b.ii.

Markscheme / solution

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

det A = 5(1) − 7(−2) = 19

A^–1 $= \frac{1}{{19}}\left( {\begin{array}{*{20}{c}} 1&2 \\ { - 7}&5 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\frac{1}{{19}}}&{\frac{2}{{19}}} \\ {\frac{{ - 7}}{{19}}}&{\frac{5}{{19}}} \end{array}} \right)$ (A2)

Note: Award (A1) for $\left( {\begin{array}{*{20}{c}} 1&2 \\ { - 7}&5 \end{array}} \right)$ , (A1) for dividing by 19.

A^–1 $= \left( {\begin{array}{*{20}{c}} {0.0526}&{0.105} \\ { - 0.368}&{0.263} \end{array}} \right)$ (G2)

[2 marks]

XA + B = C ⇒ XA = C – Β (M1)

X = (C – Β)Α^–1 (A1)

X = (C – B)A^–1 (A2)

[2 marks]

b.i.

(C – Β)Α^–1 = $\left( {\begin{array}{*{20}{c}} { - 11}&{ - 7} \\ { - 13}&9 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\frac{1}{{19}}}&{\frac{2}{{19}}} \\ {\frac{{ - 7}}{{19}}}&{\frac{5}{{19}}} \end{array}} \right)$ (A1)

⇒ X = $\left( {\begin{array}{*{20}{c}} {\frac{{38}}{{19}}}&{\frac{{ - 57}}{{19}}} \\ {\frac{{ - 76}}{{19}}}&{\frac{{19}}{{19}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 2&{ - 3} \\ { - 4}&1 \end{array}} \right)$ (A1)

X = $\left( {\begin{array}{*{20}{c}} 2&{ - 3} \\ { - 4}&1 \end{array}} \right)$ (G2)

Note: If premultiplication by A^–1 is used, award (M1)(M0) in part (i) but award (A2) for $\left( {\begin{array}{*{20}{c}} {\frac{{ - 37}}{{19}}}&{\frac{{11}}{{19}}} \\ {\frac{{12}}{{19}}}&{\frac{{94}}{{19}}} \end{array}} \right)$ in part (ii).

[2 marks]

b.ii.

Examiners’ report

[N/A]

b.i.

[N/A]

b.ii.