Let B = [ 1 3 1 2 3 4 ] , 2 be the adjoint of a matrix A and | A | = 2 . Then [ - 2 ] ? B [ - 2 ] is equal to ? [2023]

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Solution

Q.
Let $B = [\begin{matrix} 1 & 3 & α \\ 1 & 2 & 3 \\ α & α & 4 \end{matrix}], α > 2$ be the adjoint of a matrix $A$ and $| A |$ $= 2$ . Then $[α - 2 α α] B [\begin{matrix} α \\ - 2 α \\ α \end{matrix}] is equal to$ [2023]

1 0

2 16

3 32

4 - 16

Ans.
(4)

We have, $adj A = B = [\begin{matrix} 1 & 3 & α \\ 1 & 2 & 3 \\ α & α & 4 \end{matrix}], α > 2$ and $| A |$ $= 2$

$Now, [α - 2 α α] B [\begin{matrix} α \\ - 2 α \\ α \end{matrix}]$

$= [α - 2 α α] [\begin{matrix} 1 & 3 & α \\ 1 & 2 & 3 \\ α & α & 4 \end{matrix}] [\begin{matrix} α \\ - 2 α \\ α \end{matrix}]$

$= [α - 2 α α] [\begin{matrix} α - 6 α + α^{2} \\ α - 4 α + 3 α \\ α^{2} - 2 α^{2} + 4 α \end{matrix}]$

$= α (α - 6 α + α^{2}) - 2 α (α - 4 α + 3 α) + α (α^{2} - 2 α^{2} + 4 α)$

$= α^{2} - 6 α^{2} + α^{3} - 2 α^{2} + 8 α^{2} - 6 α^{2} + α^{3} - 2 α^{3} + 4 α^{2} = - α^{2}$

$Now,$ $| B |$ $=$ $| adj A |$ $= {(| A |)}^{2} = 2^{2} = 4$

$∴$ $8 - 3 α - 12 + 9 α - α^{2} = 4$

$\Rightarrow α^{2} - 6 α + 8 = 0 \Rightarrow (α - 2) (α - 4) = 0 \Rightarrow α = 2, 4$

$But α > 2$ $∴$ $α = 4$

$∴$ $[α - 2 α α] B [\begin{matrix} α \\ - 2 α \\ α \end{matrix}] = - {(4)}^{2} = - 16$

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