EXM.1.AHL.TZ0.42

pestleMathematicsAIHLPaper 1EXM· ahl-1-15-eigenvalues-and-eigenvectorssource ↗

Given the matrix A = $\left( {\begin{array}{*{20}{c}} 3&2 \\ { - 1}&0 \end{array}} \right)$ find the values of the real number $k$ for which ${\text{det}}\left( {A - kI} \right) = 0$ where $I = \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right)$

Markscheme / solution

${\text{det}}\left( {A - kI} \right) = 0$

$\Rightarrow \left| {\begin{array}{*{20}{c}} {3 - k}&2 \\ { - 1}&{ - k} \end{array}} \right| = 0$ (M1)

$\Rightarrow {k^2} - 3k + 2 = 0$ (M1)

$\Rightarrow \left( {k - 2} \right)\left( {k - 1} \right) = 0$

$\Rightarrow k = 1{\text{,}}\,\,2$ (A2) (C4)

[4 marks]

Examiners’ report

[N/A]