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3 and 4 . determinant and metrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths3 and 4 . determinant and metrices1 Mark
MCQ
If $A = \left[ {\begin{array}{*{20}{c}}0&1 \\ 1&0\end{array}} \right],$then ${A^4}$=
  • ✓
    $\left[ {\begin{array}{*{20}{c}}1&0 \\ 0&1\end{array}} \right]$
  • B
    $\left[ {\begin{array}{*{20}{c}}1&1 \\ 0&0\end{array}} \right]$
  • C
    $\left[ {\begin{array}{*{20}{c}}0&0 \\ 1&1\end{array}} \right]$
  • D
    $\left[ {\begin{array}{*{20}{c}}0&1 \\1&0\end{array}} \right]$
✓

Answer

Correct option: A.
$\left[ {\begin{array}{*{20}{c}}1&0 \\ 0&1\end{array}} \right]$
a
(a) We have $A = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]$

$\therefore$ ${A^2} = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\,\,\left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right] = {I_2}$

$\therefore$ ${A^4} = {A^2}.{A^2} = {I_2}.{I_2} = {I_2} = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right]$.

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