Let A=1 1 00 1 00 0 1, and let I denote the 3 × 3 identity matrix. Then 2A2−A3=

We have A2=110010001110010001 =120010001 A3 =A2A =130010001 ∴ 2A2-A3=A

Let A=1 1 00 1 00 0 1, and let I denote the 3 × 3 identity matrix. Then 2A2−A3=

We have A2=110010001110010001 =120010001 A3 =A2A =130010001 ∴ 2A2-A3=A

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Let $A = [\begin{array}{lll} 1 1 0 \\ 0 1 0 \\ 0 0 1 \end{array}]$ , and let $I$ denote the $3 \times 3$ identity matrix. Then $2 A^{2} - A^{3} =$

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$A - I$

$I - A$

$A + I$

$A$

answer is D.

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Detailed Solution

We have $A^{2} = (\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}) (\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}) = (\begin{matrix} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) A^{3} = A^{2} A = (\begin{matrix} 1 & 3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) ∴ 2 A^{2} - A^{3} = A$