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 \( \mathrm{A} \) is said to be involutory if \( \mathrm{A}^{2}=\mathrm{I}, \mathrm{I} \) 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 \( \quad A^{\prime} A=I=A A^{\prime} \).
If \( A \) and \( B \) are two square matrices such that \( A B=A \& B A=B \), then \( A \& B \) are
- (A) Idempotent matrices
(B) Involutory matrices
(C) Orthogonal matrices
(D) Nilpotent matrices
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