Question
Download Solution PDFIf \(A = \left[ {\begin{array}{*{20}{c}} 0&1\\ 1&0 \end{array}} \right],\) then the matrix A is a/an
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
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.Last updated on Jul 8, 2025
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