Let A be a 3 3 real matrix such that A^2(A-2 I)-4(A-I)=O, where I and O are the identity a…

Question

Let $A$ be a $3 	imes 3$ real matrix such that $A^2(A-2 I)-4(A-I)=O$, where $I$ and $O$ are the identity and null matrices, respectively. If $A^5=\alpha A^2+\beta A+\gamma I$, where $\alpha, \beta$, and $\gamma$

Accepted Answer

$\begin{gathered} {A}^{2}(A-2I)-4(A-I)=0 \ {A}^{3}-2{A}^{2}-4A+4I=0 \end{gathered}$

Multiply by $A$

$\begin{gathered} {A}^{4}=2{A}^{3}+4{A}^{2}-4A \ {A}^{4}=2(2{A}^{2}+4A-4I)+4{A}^{2}-4A \ {A}^{4}=8{A}^{2}+4A-8I \end{gathered}$

Multiply again by $A$

$\begin{gathered} \Rightarrow {A}^{5}=8{A}^{3}+4{A}^{2}-8A \ \Rightarrow {A}^{5}=8(2{A}^{2}+4A-4I)+4{A}^{2}-8A \ \Rightarrow {A}^{5}=20{A}^{2}+24A-32I \end{gathered}$

Comparing with ${A}^{5}=\alpha {A}^{2}+\beta A+\gamma I$

$\begin{gathered} \alpha =20,\beta =24,\gamma =-32 \ 	herefore \alpha +\beta +\gamma =20+24-32 \ =44-32 \ =12 \end{gathered}$

Solutions · 1