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Solve the following.
Find the value of k if M = (1 2; 2 3) and M² - kM - I₂ = 0.
Find the value of $k$ if $M = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$ and $M^2 - kM - I_2 = 0$.
Answer
Answer
$M^2 = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 5 & 8 \\ 8 & 13 \end{pmatrix}$
$M^2 - kM - I_2 = \begin{pmatrix} 4-k & 8-2k \\ 8-2k & 12-3k \end{pmatrix} = 0$
$4 - k = 0$, $8 - 2k = 0$, $12 - 3k = 0$ all give $k = 4$.
Hence $k = 4$.
Final answer: $4$