Let [A=[[arraylll 2 & a & 0 1 & 3 & 1 0 & 5 & b array]]].

Question

Let $$A=\left[\[\begin{array}{lll} 2 & a & 0 \ 1 & 3 & 1 \ 0 & 5 & b \end{array}$$\right]\]. If $A^3=4 A^2-A-21 I$, where $I$ is the identity matrix of order $$3 \times 3$$, then $2 a+3 b$ is equal to

Accepted Answer

$$\begin{aligned}
& |A-\lambda I|=0 \
& \left|\[\begin{array}{ccc} 2-\lambda & a & 0 \ 1 & 3-\lambda & 1 \ 0 & 5 & b-\lambda \end{array}\end{aligned}$$\right|=0 \
\& (2-\lambda)[(3-\lambda)(b-\lambda)-5]-a[b-\lambda-0]+0=0 \
\& (2-\lambda)\left[3 b-3 \lambda-b \lambda+\lambda\^2-5\right]-a b+a \lambda=0 \
\& \lambda\^3-(b+5) \lambda\^2+(1-a+5 b) \lambda+(10-6 b+a b)=0 \
\& A\^3-(b+5) A\^2+(1-a+5 b) A+(10-6 b+a b) I=0 \
\& \Rightarrow \mathrm{b}+5=4,1-a+5 b=1,10-6 b+a b=21 \
\& \Rightarrow a=-5, b=-1 \
\& \Rightarrow 2 a+3 b=-13
\]

Solutions · 1