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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
Let $A = \left( {\begin{array}{*{20}{c}}0&0&{ - 1}\\0&{ - 1}&0\\{ - 1}&0&0\end{array}} \right)$, the only correct statement about the matrix $A$ is
  • ✓
    ${A^2} = I$
  • B
    $A = ( - 1)\,I,$ where I is a unit matrix
  • C
    ${A^{ - 1}}$ does not exist
  • D
    $A$ is a zero matrix
✓

Answer

Correct option: A.
${A^2} = I$
a
(a) Let $A = \left( {\begin{array}{*{20}{c}}0&0&{ - 1}\\0&{ - 1}&0\\{ - 1}&0&0\end{array}} \right)$

Check by options.

$(i)$ ${A^2} = \left( {\begin{array}{*{20}{c}}0&0&{ - 1}\\0&{ - 1}&0\\{ - 1}&0&0\end{array}} \right)\,\,\left( {\begin{array}{*{20}{c}}0&0&{ - 1}\\0&{ - 1}&0\\{ - 1}&0&0\end{array}} \right)$

${A^2} = \left( {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right) = I$

$(ii)$ $( - 1)\,I = \left( {\begin{array}{*{20}{c}}{ - 1}&0&0\\0&{ - 1}&0\\0&0&{ - 1}\end{array}} \right) \ne A$.

$(iii)$ $|A| = 1 \ne 0 \Rightarrow {A^{ - 1}}$ exists.

$(iv)$ Clearly $A$, is not a zero matrix.

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