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

Question

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

Aicharya · Accepted Answer

$\begin{aligned} & A^3-4 A^2+A+21 I=0 \ & \operatorname{tr}(A)=4=5+6 \Rightarrow b=-1 \ & |A|=-21 \ & -16+a=-21 \Rightarrow a=-5 \ & 2 a+3 b=-13\end{aligned}$

Let $A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$. If $A^3=4 A^2-A-21 I$…

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