Some special square matrices are defined as follows : Nilpotent matrix : A square matrix \( A \)...
0Some special square matrices are defined as follows :
Nilpotent matrix : A square matrix \( A \) is said to be nilpotent ( of order 2 ) if, \( A^{2}=0 \). A square matrix is
\( \mathrm{P} \) said to be nilpotent of order \( p \), if \( p \) is the least positive integer such that \( A p=0 \).
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Idempotent matrix : A square matrix \( A \) is said to be idempotent if, \( A^{2}=A \).
e.g. \( \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \) is an idempotent matrix.
Involutory matrix : A square matrix \( A \) is said to be involutory if \( A^{2}=I, I \) being the identity matrix.
e.g. \( A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \) is an involutory matrix.
Orthogonal matrix: A square matrix \( A \) is said to be an orthogonal matrix if \( A^{\prime} A=I=A A^{\prime} \).
If the matrix \( \left[\begin{array}{ccc}0 & 2 \beta & \gamma \\ \alpha & \beta & -\gamma \\ \alpha & -\beta & \gamma\end{array}\right] \) is orthogonal, then
(A) \( \alpha=\pm \frac{1}{\sqrt{2}} \)
(B) \( \beta=\pm \frac{1}{\sqrt{6}} \)
(C) \( \gamma=\pm \frac{1}{\sqrt{3}} \)
(D) all of these
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