Let be a real number. Consider the matrix A = ( 0 1 2 1 - 2 3 1 - 2 ) . If A 7 - ( - 1 ) A 6 - A 5 is a singular matrix, then the value of 9 is _______. [2022]

Question

Let $β$ be a real number. Consider the matrix $A = (\begin{matrix} β & 0 & 1 \\ 2 & 1 & - 2 \\ 3 & 1 & - 2 \end{matrix}) .$

If $A^{7} - (β - 1) A^{6} - β A^{5}$ is a singular matrix, then the value of $9 β$ is _______. [2022]

Smart Achievers · Accepted Answer

(3)

$A = (\begin{matrix} β & 0 & 1 \\ 2 & 1 & - 2 \\ 3 & 1 & - 2 \end{matrix}),$ $∴$ $| A |$ $= - 1$

Given that $A^{7} - (β - 1) A^{6} - β A^{5}$ is a singular matrix.

$| A^{5} |$ $| A^{2} - (β - 1) A - β I |$ $= 0$

$\Rightarrow$ $| A^{5} |$ $| (A + I) (A - β I) |$ $= 0$

$∴$ $| A^{5} |$ $| A + I |$ $| A - β I |$ $= 0$ $[∵ | A^{5} | = | A |^{5}]$

As $| A |$ $\neq 0$ , $| A + I |$ $= 0$ $or$ $| A - β I |$ $= 0$

$\Rightarrow$ $| A + I |$ $| \begin{matrix} β + 1 & 0 & 1 \\ 2 & 2 & - 2 \\ 3 & 1 & - 1 \end{matrix} |$ $= 0$

$- 1 = 0$ $(Rejected)$ ${| A + I | \neq 0}$

$∴$ $| A - β I |$ $=$ $| \begin{matrix} 0 & 0 & 1 \\ 2 & 1 - β & - 2 \\ 3 & 1 & - 2 - β \end{matrix} |$ $= 0$

$\Rightarrow 2 - 3 (1 - β) = 0$ $\Rightarrow 2 - 3 + 3 β = 0$

$\Rightarrow β = \frac{1}{3}$

$∴$ $9 β = 3$

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