Question

Which of the following matrix$(\ne I)$ may not invertible :

• A. idempotent
• B. orthogonal
• C. involutory
• D. None of the above

Answer 1

The correct option is A idempotent

For involutory matrix, $A^2=I$ $\Rightarrow \left|A^2\right|=|I|\Rightarrow |A{|}^2=1\Rightarrow |A|=\pm 1$
For idempotent matrix, $A^2=A$
$\Rightarrow \left|A^2\right|=|A|\Rightarrow |A{|}^2=|A|\Rightarrow |A|=0,1$
For orthogonal matrix, $A{A}^T=I$
$\Rightarrow \left|A{A}^T\right|=|I|\Rightarrow |A|\left|A^T\right|=1\Rightarrow |A{|}^2=1\Rightarrow |A|=\pm 1$
Thus, in idempotent matrix, inverse dosen't exist