Determinants $3 \times 3$ matrix proofs

Question

Suppose $A$ is a $3\times3$ matrix such that $\det(A)=\frac{1}{125}$. Find $\det(5A^{−1})$.

I know that this can also be written as $\det(5/A)$

However, I am struggling to work out what $A$ is

Please help

Answer 1

You need to know two important properties of the determinant:

• $\det(A^{-1}) = \frac{1}{\det(A)}$
• $\det(\lambda A) = \lambda^n \det(A)$ where $A$ is an $n \times n$ matrix

Since you know $\det(A)$, you can then determine

$$ \det(5 A^{-1}) = 5^3 \det(A^{-1}) = 5^3 \cdot \frac{1}{\frac{1}{125}} = 5^3 \cdot 125$$

The two properties I mentioned before usually go under the name multiplicativity of determinant and multilinearity of determinant. Along with some other useful properties, they are listed here.

Answer 2

Hint:

$$\text{ det }(5A^{-1})\text{ det }(A)=\text{ det }(5A^{-1}A)=\text{ det }(5I).$$