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Adjoint of a Matrix

If α h+β g ...

Question

If $∣ ∣ ∣ ∣ ∣ \begin{matrix} α h + β g & g & α β + h γ α β + β f & f & h β + β γ α f + β γ & γ & g β + f γ \end{matrix} ∣ ∣ ∣ ∣ ∣ = λ ∣ ∣ ∣ ∣ ∣ \begin{matrix} α h + β g & α & h α β + β f & h & β α f + β γ & g & f \end{matrix} ∣ ∣ ∣ ∣ ∣, t h e n λ i s$

A

α

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B

β

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C

γ

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D

2

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Solution

The correct option is B $α$
$∣ ∣ ∣ ∣ ∣ \begin{matrix} α h + β g & g & α β + h γ α β + β f & f & h β + β γ α f + β γ & γ & g β + f γ \end{matrix} ∣ ∣ ∣ ∣ ∣ = λ ∣ ∣ ∣ ∣ ∣ \begin{matrix} α h + β g & α & h α β + β f & h & β α f + β γ & g & f \end{matrix} ∣ ∣ ∣ ∣ ∣$
$L H S = ∣ ∣ ∣ ∣ ∣ \begin{matrix} α h + & β g & g & α β α β + & β f & f & h β α f + & β γ & γ & g β \end{matrix} ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} α h + & β g & g & h γ α β + & β f & f & β γ α f + & β γ & γ & f γ \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} α h + & β g & g & α β α β + & β f & f & h β α f + & β γ & γ & g β \end{matrix} ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} α h & g & h γ α β & f & β γ α f & γ & f γ \end{matrix} ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} β g & g & h γ β f & f & β γ β γ & γ & f γ \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} α h + & β g & g & α β α β + & β f & f & h β α f + & β γ & γ & g β \end{matrix} ∣ ∣ ∣ ∣ ∣ + α γ ∣ ∣ ∣ ∣ ∣ \begin{matrix} h & g & h β & f & β f & γ & f \end{matrix} ∣ ∣ ∣ ∣ ∣ + β ∣ ∣ ∣ ∣ ∣ \begin{matrix} g & g & h γ f & f & β γ γ & γ & f γ \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} α h + & β g & g & α β α β + & β f & f & h β α f + & β γ & γ & g β \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= α ∣ ∣ ∣ ∣ ∣ \begin{matrix} h & g & α β β & f & h β f & γ & g β \end{matrix} ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} β g & g & α β β f & f & h β β γ & γ & g β \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= α ∣ ∣ ∣ ∣ ∣ \begin{matrix} h & g & α β β & f & h β f & γ & g β \end{matrix} ∣ ∣ ∣ ∣ ∣ + 0$
$= α β ∣ ∣ ∣ ∣ ∣ \begin{matrix} h & g & α β & f & h f & γ & g \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= α ∣ ∣ ∣ ∣ ∣ \begin{matrix} h & β g & α β & β f & h f & β γ & g \end{matrix} ∣ ∣ ∣ ∣ ∣$
$C_{2} \to C_{2} + α C_{1}$
$= α ∣ ∣ ∣ ∣ ∣ \begin{matrix} h & α h + β g & α β & α β + β f & h f & α f + β γ & g \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= α ∣ ∣ ∣ ∣ ∣ \begin{matrix} α h + β g & α & h α β + β f & h & β α f + β γ & g & f \end{matrix} ∣ ∣ ∣ ∣ ∣$
So, comparing with RHS of given expression
$λ = α$

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