Question

If A is unimodular, then which of the following is unimodular?

• A. -A
• B. A${}^{-1}$
• C. adj A
• D. $\omega$A, where $\omega$ is cube root of unity

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A -A
B A${}^{-1}$
C $\omega$A, where $\omega$ is cube root of unity
D adj A

Given $A$ is unimodular i.e., $|A|=\pm 1$

(A) $|-A|=(-1{)}^n|A|=(-1{)}^n\times (\pm 1)=\pm 1$

It is unimodular.

(B) $|A^{-1}|=|adj\left(A\right)|\frac{1}{|A|}=|A{|}^{n-2}=(\pm 1{)}^{n-2}=\pm 1$

It is unimodular.

(C) $|Adj\left(A\right)|=|A{|}^{n-1}=(\pm 1{)}^{n-1}$

if $n$ is even, $|Adj\left(A\right)|=\pm 1$

else if $n$ is odd, $|Adj\left(A\right)|=1$

It is unimodular.

(D) $|\omega A|=(\omega {)}^n|A|$

$\omega$ is a complaex number whose modulus is 1. Also modulus of ${\omega }^n$ is 1

Hence, the correct options are$\left(A\right),\left(B\right),\left(C\right)$ and $\left(D\right)$