Question

If $A$ is an invertible matrix of order $2$, then $det\left(A^{-1}\right)$ is equal to

• A. $det\left(A\right)$
• B. $\frac{1}{det\left(A\right)}$
• C. $1$
• D. $0$

Answer 1

The correct option is B $\frac{1}{det\left(A\right)}$

We know that

$A{A}^{-1}=I$

Taking determinant both sides

$|A{A}^{-1}|=|I|$

$|A||A^{-1}|=|I|$ $[\because |AB|=|A||B|]$

$|A||A^{-1}|=1$ $[\because |I|=1]$

$|A^{-1}|=\frac{1}{|A|}$

Since $|A|\ne 0$

Hence, $|A^{-1}|=\frac{1}{|A|}$