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

Options

A. idempotent
B. orthogonal
C. involuntary
D. none of these

Answer

A

Explanation

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.