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. $\frac{1}{\sqrt[5]{5625}}$
• D. $\pm 0.2$

Answer 1

The correct option is D $\pm 0.2$

$A^{-1}=\frac{1}{detA}$ (adjoint of $A$)

Taking determinant on both sides, we can write

$det\left(A^{-1}\right)\times (det(A){)}^3=det$ (adjoint of $A$) since the matrix A is of order 3.

Also, $det\left(A^{-1}\right)=\frac{1}{detA}$

$\Rightarrow {\left(\frac{1}{det\left(A^{-1}\right)}\right)}^2=25$

$\Rightarrow det\left(A^{-1}\right)=\frac{1}{\sqrt{25}}=\pm 0.2$