If P = 1 α 3 1 3 3 2 4 4 is the adjoint of a 3 × 3 matrix A and A = 4 , then α is equal to

Given P = a d j A ⇒ P = a d j A = A 2 ∵ a d j A = A n - 1 where n = 3 Now, P = 1 α 3 1 3 3 2 4 4 = 1 × 12 − 12 − α × 4 − 6 + 3 × 4 − 6 = 2 α - 6 ⇒ 2 α - 6 = 1 6 ⇒ α = 11

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MathematicsMatricesAdjointMedium2 minPYQ_2013

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If $P = [\begin{matrix} 1 & α & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{matrix}]$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A| = 4$ , then $α$ is equal to

Options:

Answer:

Solution:

Given $P = (a d j A)$

$\Rightarrow |P| = |a d j A| = {|A|}^{2}$ $(∵ |a d j A| = {|A|}^{n - 1} where n = 3)$

Now, $|P| = |\begin{matrix} 1 & α & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{matrix}| = 1 \times (12 - 12) - α \times (4 - 6) + 3 \times (4 - 6) = 2 α - 6$

$\Rightarrow 2 α - 6 = 16$

$\Rightarrow$ $α = 11$

Stream:JEESubject:MathematicsTopic:MatricesSubtopic:Adjoint

⏱ 2mℹ️ Source: PYQ_2013

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