If A=1 −1 11 −1 11 −1 1 then, A5−A4−A3+A2 is equal to

A2=1−111−111−111−111−111−11=1−1+1−1+1−11−1+11−1+1−1+1−11−1+11−1+1−1+1−11−1+1=1−111−111−11=AA3=A2.A=A2=ASimilarly, A4=A and A5=ATherefore, A5−A4−A3+A2=A−A−A+A=O

If A=1 −1 11 −1 11 −1 1 then, A5−A4−A3+A2 is equal to

A2=1−111−111−111−111−111−11=1−1+1−1+1−11−1+11−1+1−1+1−11−1+11−1+1−1+1−11−1+1=1−111−111−11=AA3=A2.A=A2=ASimilarly, A4=A and A5=ATherefore, A5−A4−A3+A2=A−A−A+A=O

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If $A = [\begin{array}{lll} 1 - 1 1 \\ 1 - 1 1 \\ 1 - 1 1 \end{array}]$ then, $A^{5} - A^{4} - A^{3} + A^{2}$ is equal to

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Detailed Solution

$\begin{array}{l} A^{2} = [\begin{array}{lll} 1 & - 1 & 1 \\ 1 & - 1 & 1 \\ 1 & - 1 & 1 \end{array}] [\begin{array}{lll} 1 & - 1 & 1 \\ 1 & - 1 & 1 \\ 1 & - 1 & 1 \end{array}]⌋ = [\begin{array}{lll} 1 - 1 + 1 & - 1 + 1 - 1 & 1 - 1 + 1 \\ 1 - 1 + 1 & - 1 + 1 - 1 & 1 - 1 + 1 \\ 1 - 1 + 1 & - 1 + 1 - 1 & 1 - 1 + 1 \end{array}] = [\begin{array}{lll} 1 & - 1 & 1 \\ 1 & - 1 & 1 \\ 1 & - 1 & 1 \end{array}] = A \\ A^{3} = A^{2} . A = A^{2} = A \end{array}$
Similarly, $A^{4} = A and A^{5} = A$
Therefore, $A^{5} - A^{4} - A^{3} + A^{2} = A - A - A + A = O$