Question

Let $A$ be an orthogonal non-singular matrix of order $n$, then $|A-I_n|$ is equal to

• A. $|I_n-A|$
• B. $|A||I_n-A^T|$
• C. $|A|$
• D. $(-1{)}^n|A||I_n-A|$

Answer 1

Answer: (B)

$|A||I_n-A^T|$

Given $A$ is an orthogonal matrix then $A{A}^T=A^{T}A=I_n$......(1)

Now,

$|A-I_n|$

$=|A-A{A}^T|$ [ Using (1)]

$=|A(I_n-A^T)|$

$=|A||(I_n-A^T)|$. [ Since $|AB|=|A||B|$]