Let $P=\left[a_{i j}\right]$ be a $3 \times 3$ matrix and let $Q=\left[b_{i j}\right]$ where $b_{i j}=2^{i+j} a_{i j} ~\forall ~1 ≤ i, j ≥ 3$. If the determinant of P is 2, then the determinant of Q is:

$2^{13}$

The correct answer is Option (1) → $2^{13}$

$|P|=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}=2$

$Q=\begin{vmatrix}2^2a_{11}&2^3a_{12}&2^4a_{13}\\2^3a_{21}&2^4a_{22}&2^5a_{23}\\2^4a_{31}&2^5a_{32}&2^6a_{33}\end{vmatrix}=2$

$⇒Q=DPD$  [D → Diagonal Matrix]

$D=\begin{bmatrix} 2^1 & 0 & 0 \\ 0 & 2^2 & 0 \\ 0 & 0 & 2^3 \end{bmatrix}⇒|D|=2^1.2^2.2^3=2^6$

$∴|Q|=|DPD|$

$=|D|^2|P|$

$=(2^6)^2(2)=2^{13}$