EXM.1.AHL.TZ0.9

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

The matrix A = $\left( {\begin{array}{*{20}{c}} 1&2&0 \\ { - 3}&1&{ - 1} \\ 2&{ - 2}&1 \end{array}} \right)$ has inverse A⁻¹ = $\left( {\begin{array}{*{20}{c}} { - 1}&{ - 2}&{ - 2} \\ 3&1&1 \\ a&6&b \end{array}} \right)$ .

Consider the simultaneous equations

$x + 2y = 7$

$- 3x + y - z = 10$

$2x - 2y + z = - 12$

Write down the value of $a$ .

[1]

a.i.

Write down the value of $b$ .

[1]

a.ii.

Write these equations as a matrix equation.

[1]

Solve the matrix equation.

[3]

Markscheme / solution

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

$a$ = 4 A1 N1

[1 mark]

a.i.

$b$ = 7 A1 N1

[1 mark]

a.ii.

EITHER

A $\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 7 \\ {10} \\ { - 12} \end{array}} \right)$ A1 N1

$\left( {\begin{array}{*{20}{c}} 1&2&0 \\ { - 3}&1&{ - 1} \\ 2&{ - 2}&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 7 \\ {10} \\ { - 12} \end{array}} \right)$ A1 N1

[1 mark]

$\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right)$ = A⁻¹ $\left( {\begin{array}{*{20}{c}} 7 \\ {10} \\ { - 12} \end{array}} \right)$ (accept algebraic method) (M1)

$\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 3} \\ 5 \\ 4 \end{array}} \right)$ (accept $x$ = −3, $y$ = 5, $z$ = 4) A2 N3

[3 marks]

Examiners’ report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]