Question

Let A be an orthogonal non-singular matrix of order $n$, then the determinant of matrix $A{I}_n$ 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)

$A{A}^T=A^{T}A=I$
$|A|\ne 0$ order $n$
$\left|A-I_n\right|=?$
$A{A}^T=I_n$
$\Rightarrow A-I_n=A-A{A}^T$
$=A\left(I_n-A^T\right)$
$\Rightarrow \left|A-I_n\right|=\left|A\left(I_n-A^T\right)\right|$
$\Rightarrow |A|\left(I_n-A^T\right)$
$\Rightarrow |A|I_n-A$