In which of the following type of matrix inverse does not exist always? a. idempotent b. orthogonal c. involuntary d. none of these

Monjur Alahi

In which of the following type of matrix inverse does not exist always? a. idempotent b. orthogonal c. involuntary d. none of these
A. idempotent
B. orthogonal
C. involuntary
D. none of these


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Salim Ahmed Kawsar

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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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