Let P = [ p i j ] and Q = [ q i j ] be two square matrices of order 3 such that q i j = 2 ( i + j - 1 ) p i j and det ( Q ) = 2 10 . Then the value of det ( adj ( adj P ) ) is: [2026]

Question

Let $P = [p_{i j}]$ and $Q = [q_{i j}]$ be two square matrices of order 3 such that $q_{i j} = 2^{(i + j - 1)} p_{i j}$ and $\det (Q) = 2^{10} .$ Then the value of $\det (adj (adj P))$ is: [2026]

Smart Achievers · Accepted Answer

(2)

$| \begin{matrix} 2 p_{11} & 2^{2} p_{12} & 2^{3} p_{13} \\ 2^{2} p_{21} & 2^{3} p_{22} & 2^{4} p_{23} \\ 2^{3} p_{31} & 2^{4} p_{32} & 2^{5} p_{33} \end{matrix} |$ $= 2^{10}$

$= 2^{2} \cdot 2 \cdot 2^{3}$ $| \begin{matrix} p_{11} & p_{12} & p_{13} \\ 2 p_{21} & 2 p_{22} & 2 p_{23} \\ 2^{2} p_{31} & 2^{2} p_{32} & 2^{2} p_{33} \end{matrix} |$ $= 2^{10}$

$= 2^{9}$ $| \begin{matrix} p_{11} & p_{12} & p_{13} \\ p_{21} & p_{22} & p_{23} \\ p_{31} & p_{32} & p_{33} \end{matrix} |$ $= 2^{10}$ $\Rightarrow$ $| P |$ $= 2$

$| adj (adj (P)) |$ $=$ $| P |^{{(n - 1)}^{2}}$ $=$ $| P |^{4} = 2^{4} = 16$

Solution

Q.
Let $P = [p_{i j}]$ and $Q = [q_{i j}]$ be two square matrices of order 3 such that $q_{i j} = 2^{(i + j - 1)} p_{i j}$ and $\det (Q) = 2^{10} .$ Then the value of $\det (adj (adj P))$ is: [2026]

1 124

2 16

3 81

4 32

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