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)

$A \, I_n - A$

Answer 2

The correct option is (B): $A \, I_n - A$.
$AA ^{ T }= A ^{ T } A = I$
$| A | \neq0$ order $n$
$\left| A - I _{ n }\right|=?$
$AA ^{ T }= I _{ n }$
$\Rightarrow A - I _{ n }= A - AA ^{ T }$
$= A \left( I _{ n }- A ^{ T }\right)$
$\Rightarrow\left| A - I _{ n }\right|=\left| A \left( I _{ n }- A ^{ T }\right)\right|$
$\Rightarrow| A |\left( I _{ n }- A ^{ T }\right)$
$\Rightarrow| A | I _{ n }- A$

So, the correct option is (B): $A \, I_n - A$