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In which of the following type of matrix inverse does not exist always?a. idempotent b. orthogonalc. involuntary d. none of theseA. idempotentB. orthogonalC. involuntaryD. none of these

Answer» Correct Answer - A
For involuntary matrix,
`A^(1)=I`
`implies |A^(2)|=|I|implies |A|^(2)=1 implies |A|= pm 1`
For idempotent matrix,
`A^(2)=A`
`implies |A^(2)|=|A|implies |A|^(2)=|A|=0` or 1
for orthogonal matrix,
`A A^(T)=I`
`implies |AA^(T)|=|I|implies|A||A^(T)|=1implies |A|^(2)=1 implies |A|= pm 1`
Thus, if matrix A is idempotent it may not be invertible.


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