If \(A = \left[ {\begin{array}{*{20}{c}} 0&1\\ 1&0 \end{array}} \right],\) then the matrix A is a/an

Concept:

Singular Matrix: Any square matrix of order n is said to be singular if |A| = 0.

Involuntary Matrix: Any square matrix of order n is said to be an involuntary matrix if A² = I, where I is the identity matrix of order n.

Nilpotent Matrix: Any square matrix of order n is said to be nilpotent matrix if there exist least positive integer m such that A^m = O, where O is the null matrix of order n.

Idempotent Matrix: Any square matrix of order n is said to be an idempotent matrix if A² = A.

Calculation:

Given: \(A = \left[ {\begin{array}{*{20}{c}} 0&1\\ 1&0 \end{array}} \right],\)

\(\Rightarrow \;{A^2} = \;\left[ {\begin{array}{*{20}{c}} 0&1\\ 1&0 \end{array}} \right] \times \;\left[ {\begin{array}{*{20}{c}} 0&1\\ 1&0 \end{array}} \right] \\= \;\left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right] = I\)