Question

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

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

Answer 1

The correct option is A idempotent

For involuntary matrix ,

$A^2=I$

$\Rightarrow |A^2|=|i|\Rightarrow |A^2|=1\Rightarrow |A|=\pm 2$ for

idempotent matrix,

$A^2=A$

$\Rightarrow |A^2|=|A|\Rightarrow |A{|}^2=|A|\Rightarrow |A|=0or1$

For orthogonal matrix,

$A{A}^T=I$

$\Rightarrow |A{A}^T|=|I|\Rightarrow |A||A^T|=1\Rightarrow |A{|}^2=1\Rightarrow |A|=\pm 1$ thus if matrix A is idempotent it may not be invertible.