Question

The characteristic equation of a matrix $A$ is ${\Omega }^3-5{\Omega }^2-3\Omega +2=0$ then $|\text{adj}\left(A\right)|=$

• A. $9$
• B. $25$
• C. $\frac{1}{2}$
• D. $4$

Answer 1

The correct option is D $4$

The characteristic equation of matrix A is ${\Omega }^3-5{\Omega }^2-3\Omega +2=0$

$\Rightarrow |\Omega I-A|={\Omega }^3-5{\Omega }^2-3\Omega +2$

Now put $\Omega =0$

$\Rightarrow |-A|=|A|=2$

The given matix $A$ is $3\times 3$

$|\text{adj}\left(A\right)|=|A{|}^{3-1}=2^2=4$