18M.1.AHL.TZ0.F_2

pestleMathematicsAIHLPaper 118M· ahl-1-14-introduction-to-matricessource ↗

Let A² = 2A + I where A is a 2 × 2 matrix.

Show that A⁴ = 12A + 5I.

[3]

Let B = $\left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&{ - 3} \end{array}} \right]$ .

Given that B² – B – 4I = $\left[ {\begin{array}{*{20}{c}} k&0 \\ 0&k \end{array}} \right]$ , find the value of $k$ .

[3]

Markscheme / solution

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

METHOD 1
A⁴ = 4A² + 4AI + I² or equivalent M1A1
= 4(2A + I) + 4A + I A1
= 8A + 4I + 4A + I
= 12A + 5I AG

[3 marks]

METHOD 2
A³ = A(2A + I) = 2A² + AI = 2(2A + I) + A(= 5A + 2I) M1A1
A⁴ = A(5A + 2I) A1
= 5A² + 2A = 5(2A + I) + 2A
= 12A + 5I AG

[3 marks]

B² = $\left[ {\begin{array}{*{20}{c}} {18}&2 \\ 1&{11} \end{array}} \right]$ (A1)

$\left[ {\begin{array}{*{20}{c}} {18}&2 \\ 1&{11} \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&{ - 3} \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} 4&0 \\ 0&4 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {10}&0 \\ 0&{10} \end{array}} \right]$ (A1)

$\Rightarrow k = 10$ A1

[3 marks]

Examiners’ report

[N/A]