Question

If $A$ is a square matrix of order $3$ and det $A=5$, then what is det $\left[\right(2A{)}^{-1}]$ equal to?

• A. $\frac{1}{10}$
• B. $\frac{2}{5}$
• C. $\frac{8}{5}$
• D. $\frac{1}{40}$

Answer 1

The correct option is D $\frac{1}{40}$

If $A$ is of order $3$ then,$A^{-}1$ is also of order $3$.

Now, $\text{det}\left(cA\right)=c^n\left(\text{det}A\right)$ where $n$ is the order of the matrix.

And, $\text{det}A=\frac{1}{\text{det}{A}^{-1}}$

Thus $\text{det}\left[\right(2A{)}^{-1}]=2^3\text{det}[A]$

and $\text{det}\left[A^{-1}\right]=\frac{1}{8.5}$

$=\frac{1}{40}$