In which of the following type of matrix inverse does not exist always

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idempotent

orthogonal

involutary

none of these

answer is A.

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Detailed Solution

For involutary matrix, A2=I⇒A2=|I|⇒|A|2=1⇒|A|=±1 For idempotent matrix,A2=A⇒A2=|A|⇒|A|2=|A|⇒|A|=0 or 1 For orthogonal matrix, AAT=I⇒AAT=|I|⇒A∥AT=1⇒|A|2=1⇒|A|=±1 Thus if matrix A is idempotent it may not be invertible.

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