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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
The matrix $A = \left[ {\begin{array}{*{20}{c}}{1/\sqrt 2 }&{1/\sqrt 2 }\\{ - 1/\sqrt 2 }&{ - 1/\sqrt 2 }\end{array}} \right]$ is
  • A
    Unitary
  • B
    Orthogonal
  • ✓
    Nilpotent
  • D
    Involutory
✓

Answer

Correct option: C.
Nilpotent
c
(c) ${A^2} = A\,.\,A = \left[ {\begin{array}{*{20}{c}}{1/\sqrt 2 }&{1/\sqrt 2 }\\{ - 1/\sqrt 2 }&{ - 1/\sqrt 2 }\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}{1/\sqrt 2 }&{1/\sqrt 2 }\\{ - 1/\sqrt 2 }&{ - 1/\sqrt 2 }\end{array}} \right]$

$ = \left[ {\begin{array}{*{20}{c}}0&0\\0&0\end{array}} \right] = O$

$\therefore $ Matrix $A$ is nilpotent of order  $2$ .

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