Question

Let A be an orthogonal non-singular matrix of order $n$, then the determinant of matrix $AI _{ n }$ ie, $\left| A - I _{ n }\right|$ is equal to

• A. $I_n - A$
• B. $A \, I_n - A $
• C. $A $
• D. $ -1^n \, A \, I_n - A $

Answer 1

Answer: (B)

$AA ^{ T }= A ^{ T } A = I $
$| A | \neq0$ order $n$
$\left| A - I _{ n }\right|=?$
$AA ^{ T }= I _{ n }$
$\Rightarrow A - I _{ n }= A - AA ^{ T }$
$= A \left( I _{ n }- A ^{ T }\right)$
$\Rightarrow \left| A - I _{ n }\right|=\left| A \left( I _{ n }- A ^{ T }\right)\right|$
$\Rightarrow | A |\left( I _{ n }- A ^{ T }\right)$
$\Rightarrow | A | I _{ n }- A$