Question

The characteristic equation of a matrix $A$ is $\lambda^3 - 5\lambda^2 - 3\lambda + 2 = 0$ then $| adj (A) |$ is equal to

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

Answer 1

Answer: (D)

$4$

Answer 2

We know that, $\begin{bmatrix}0&0&-c\\\ 1&0&-b\\\ 0&1&-a\end{bmatrix}$iff $\lambda^{3}+a\lambda^{2}+b\lambda+c=0$ But given that , $\lambda^{3}-5\lambda^{2}-3\lambda+2=0$ $\therefore A=\begin{bmatrix}0&0&-2\\\ 1&0&3\\\ 0&1&5\end{bmatrix}$ Now, $\left|A\right|=\begin{bmatrix}0&0&-2\\\ 1&2&3\\\ 0&1&5\end{bmatrix}$ $= 0 + 0 - 2 (1 - 0) = - 2$ $\therefore \left|adj\, A \left|=\right| A^{2} = \left(- 2\right)^{2} = 4\right|$