matrix powers problem

Question

let $ A $ be the matrix :\begin{bmatrix}1 & 3 & 1\\4 & 2 & 3\\2 & 1 & 1 \end{bmatrix}

Prove that $A$ verifies the expression :

$ -A^{3}+4A^{2}+12A+5 I_{3} = O_{3}$

Deduct that $A$ is invertible n calculate it . really need some hints

Answer 1

I hope you can do the calculation: $$A^3=\{\{77,76,56\},\{120,105,88\},\{56,48,41\}\}\\ A^2=\{\{15,10,11\},\{18,19,13\},\{8,9,6\}\}\\ A=\{\{1,3,1\},\{4,2,3\},\{2,1,1\}\}$$ Then try putting them in given equation.Then premultiply the equation by $A^{-1}$

Answer 2

Hint: find the characteristic polynomial by solving $$|A-\lambda I_3|=0,$$ where $$I_3=\begin{bmatrix} 1&0&0 \\0&1&0\\0&0&1\end{bmatrix}.$$

Now use the Cayley-Hamilton theorem (which says that every square matrix satisfies its own characteristic polynomial) (i.e. replace all the $\lambda$s in the characteristic polynomial with $A$s).

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For the second part, use the fact that $$-A^{3}+4A^{2}+12 A+5 I_{3} = 0_{3} \iff \color{green}{I_3=\frac{1}{5}\left[A^3-4A^2-12A\right]}.$$

Now what happens to $\color{green}{\text{this}}$ equation if we (pre-)multiply everything by $A^{-1} \quad$?