If A=2−2−4−1341−2−k is an idempotent matrix then k=

Solution:

$A^{2} = A (\begin{array}{ccc} 2 & - 2 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 2 & - k \end{array}) (\begin{array}{ccc} 2 & - 2 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 2 & - k \end{array}) = (\begin{array}{ccc} 2 & - 2 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 2 & - k \end{array}) (\begin{array}{ccc} 2 & - 2 & - 16 + 4 k \\ - 1 & 3 & 16 - 4 k \\ 1 & - 2 & - 12 + k^{2} \end{array}) = (\begin{array}{ccc} 2 & - 2 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 2 & - k \end{array}) - 16 + 4 k = - 4 \Rightarrow 4 k = 12 \Rightarrow k = 3$

$If A = (\begin{array}{ccc} 2 & - 2 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 2 & - k \end{array}) i s a n i d e m p o t e n t m a t r i x t h e n k =$

Solution:

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