Finding 3x3 matrix from whose given characteristic polynomial

Question

I have a simple question but I was not able to solve it. Question is this:

Find a 3x3 matrix whose charactestic polynomial is $\lambda\ ^3 - 2\lambda\ ^2 + \lambda\ + 5 $.

Can you help me how can I solve this problem?

Answer 1

In general, the $n \times n$ Companion Matrix $$\begin{bmatrix} 0 & 0 & \dots & 0 & -a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -a_{n-1} \end{bmatrix}$$ has the characteristic polynomial $a_0+a_1\lambda+\cdots+a_{n-1}\lambda^{n-1}+\lambda^n$.

Now, simply apply this fact for $n = 3$ and $a_0 = 5$, $a_1 = 1$, $a_2 = -2$, to get a $3 \times 3$ matrix whose characteristic polynomial is $5+\lambda-2\lambda^2+\lambda^3$.