Question

Let A be an orthogonal non-singular matrix of order n, then the determinant of matrix $A{I}_n$ ie, $|A-I_n|$ is equal to

• A. $I_n-A$
• B. $A{I}_n-A$
• C. $A$
• D. $-1^{n}A{I}_n-A$

Answer 1

The correct option is B $A{I}_n-A$

$A{A}^T=A^{T}A=I\left|A\right|\ne 0$ order n

$|A-I_n|=?$

$A{A}^T=I_n$

$\Rightarrow A-I_n=A-A{A}^T$

$=A(I_n-A^T)$

$\Rightarrow |A-I_n|=\left|A(I_n-A^T)\right|$

$\Rightarrow \left|A\right|(I_n-A^T)$

$\Rightarrow \left|A\right|I_n-A$

Option B