Let A = [ 2 a 0 1 3 1 0 5 b ] . If A 3 = 4 A 2 - A - 21 I , where I is the identity matrix of order 3 × 3 , then 2 a + 3 b is equal to [2024]

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Solution

Q.
Let $A = [\begin{matrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{matrix}] .$ 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 [2024]

1 $- 12$

2 $- 13$

3 $- 10$

4 $- 9$

Ans.
(2)

$A = [\begin{matrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{matrix}],$ $A^{3} = 4 A^{2} - A - 21 I$

characteristic equation of $A$ is given by

$x^{3} - 4 x^{2} + x + 21 = 0$

Sum of roots = 4 = trace of A

$\Rightarrow 2 + 3 + b = 4 \Rightarrow b = - 1$

Also, product of roots = $- 21$ = det A

$\Rightarrow 2 (3 b - 5) - a (b) = - 21 \Rightarrow - 6 - 10 + a = - 21$

$\Rightarrow a = - 21 + 16 = - 5$ $∴$ $2 a + 3 b = - 10 - 3 = - 13$

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