Question

If A is a square matrix of order $3$ such that $A{A}^T=I$ and
$\left|A\right|=1$, then $|A-I|=$

• A. $0$
• B. $1$
• C. $-1$
• D. $noneofthese$

Answer 1

The correct option is A $0$

$A$ is a square matrix of order 3 such that $A{A}^T=Iand|A|=1$
$A^T=A^{-1}\Rightarrow A^{T}A=I$
$(A-I{)}^T=A^T-I^T=A^T-I$
$\to A^T-A^{T}A=A^T(I-A)$
taking dterminant on both sides,
$|(A-I{)}^T|=|A^T(I-A)|$
$\Rightarrow |(A-I)|=|A||I-A|$
$\Rightarrow |(A-I)|=-|(A-I)|$
$\therefore |A-I|=0$