Question

If the determinant of the adjoint of a (real) matrix of order $3$ is $25$, then the determinant of the inverse of the matrix is

• A. $0.2$
• B. $\pm$ 5
• C. $\frac{1}{\sqrt[5]{625}}$
• D. $\pm 0.2$

Answer 1

Answer: (D)

Given , the determinant of the adjoint of a (real) matrix of order $3$ is $25$.
i.e., $| $ adj $ A|=25\,\,\,\,\,\,...(i)$
We know that,
$| $ adj $ A|=|A|^{n-1} \,\,\,\,\, ($ here, $\left.n=3\right)$
$\Rightarrow |A|^{3-1}=|A|^{2}=25 \,\,\,\,\,[$ from Eq. (i) $]$
$\Rightarrow |A|=\pm 5$
$\therefore \left|A^{-1}\right|=|A|^{-1}=\frac{1}{|A|}=\pm \frac{1}{5}=\pm 0.2\,\,\,\,$(by property)