Let A = [(2 a 0), (1 3 1), (2 5 b)]. If A^3 = 4A^2 - A - 21I, where I is the identity matrix of order 3 x 3 then 2a + 3b is equal to :

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 2a + 3b is equal to :

(1) –10 

(2) –13 

(3) –9 

(4) –12

Answer 1

Correct option is (2) –13 

\(A^{3}-4 A^{2}+A+21 I=0\)

\(\operatorname{tr}(\mathrm{A})=4=5+6 \Rightarrow \mathrm{b}=-1\)

\(|\mathrm{A}|=-21\)

\(-16+a=-21 \Rightarrow a=-5\)

\(2 a+3 b=-13\)