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

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NDA (Held On: 21 Apr 2019) Maths Previous Year paper
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  1. Singular matrix
  2. involuntary matrix
  3. Nilpotent matrix
  4. Idempotent matrix

Answer (Detailed Solution Below)

Option 2 : involuntary matrix
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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 Am = 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: \(A = \left[ {\begin{array}{*{20}{c}} 0&1\\ 1&0 \end{array}} \right],\)

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

Hence, A is an involuntary matrix as A2 = I.
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