Let A=[[|gathered 2 a 0 1 3 1 0 5 b gathered]]|.

Question

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

Accepted Answer

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

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