Let A=[array|c|c| & -1 6 & array], 0, such that (A)=0 and + =1.

Question

Let $A=[\begin{array}{|c|c|} \alpha  & -1 \ 6 & \beta \end{array}],\alpha >0$, such that $\det (A)=0$ and $\alpha +\beta =1$. If I denotes $2	imes 2$ identity matrix, then the matrix $(I+A{)}^{8}$ is :

Accepted Answer

Let $|A|=0 \Rightarrow \alpha \beta-(-6)=0 \Rightarrow \alpha \beta=-6$ and $\alpha+\beta=1 \Rightarrow \alpha \beta$ are roots of the equation

$$\begin{aligned}
& x^2-x-6=0 \Rightarrow x=3,-2 . \text { Since } \alpha>0 \
& \Rightarrow \quad \alpha=3, \beta=-2 \
& \Rightarrow \quad A=\left[\[\begin{array}{ll} 3 & -1 \ 6 & -2 \end{array}\end{aligned}$$\right] \Rightarrow I+A=\left[$$\begin{array}{ll} 4 & -1 \ 6 & -1 \end{array}$$\right] \
\& (I+A)\^2=\left[$$\begin{array}{ll} 10 & -3 \ 18 & -5 \end{array}$$\right] \Rightarrow(I+A)\^4=\left[$$\begin{array}{ll} 46 & -15 \ 90 & -29 \end{array}$$\right] \
\& \Rightarrow \quad(I+A)\^8=\left[$$\begin{array}{cc} 766 & -255 \ 1530 & -509 \end{array}$$\right]
\]

Solutions · 1