Question

If A is a 3x3 Matrix such that $|5 \cdot \text{adj}(A)| = 5, \text{ then } |A|$ is equal to

• A. $\pm 1$
• B. $\pm \frac{1}{5}$
• C. $\pm \frac{1}{25}$
• D. $\pm 5$

Answer 1

Answer: (B)

$\pm \frac{1}{5}$

Answer 2

The property states that for any square matrix A, $|\text{adj}(A)| = |A|^{n-1}$, where n is the order of matrix A.
In this case, A is a 3x3 matrix.
So, $|\text{adj}(A)| = |A|^{3-1} = |A|^2$

From the given equation, $|5 \cdot \text{adj}(A)| = 5$, we can substitute$|adj A|$with $|A|^2 : |5A|^2 = 5$
Taking the square root of both sides, we have: $|5A| = \pm \sqrt{5}$
Now, we know that $|cA| = c^n |A|,$ where c is a constant and n is the order of matrix A.
In this case, n = 3,
so we can write:
$5^n |A| = \pm \sqrt{5} \cdot 5^3 |A|$
=$\pm \sqrt{5} \cdot 125 |A|$
= $\pm \sqrt{5} \cdot |A|$
= $\pm \frac{\sqrt{5}}{125}.$
Simplifying the expression, we have:
$|A| = \pm \sqrt{\frac{1}{25}} \quad \text{and} \quad |A| = \pm \frac{1}{5}$
Therefore, $|A|$ is equal to $\pm \frac{1}{5}$ (option B).