Question

A square matrix $A$ of size $n\times n$ is said to be unimodular if $det\left(A\right)=\pm 1$.

If $A$ is a matrix such that $A^{\prime }A=I$, then

• A. $A$ is unimodular
• B. $-A$ is unimodular
• C. $I+A$ is unimodular
• D. none of these

Answer 1

Answer: (A)

$A$ is unimodular

Given : $A{A}^{\prime }=I$

$⟹det\left(A^{\prime }A\right)=det\left(I_n\right)=1$

$⟹\left(det\right(A){)}^2=1$

$⟹detA=\pm 1$

Hence, option D is correct.