Let `P=[a_("ij")]` be a `3xx3` matrix and let `Q=[b_("ij")]`, where `b_("ij")=2^(i+j) a_("ij")` for `1 le i, j le 3`. If the determinant of P is 2, then the determinant of the matrix Q is

Options

A. `2^(10)`
B. `2^(11)`
C. `2^(12)`
D. `2^(13)`

Explanation

To find the determinant of the matrix \( Q \) given the matrix \( P \) and the relationship between their elements, we can follow these steps:

### Step 1: Define the matrices
Let \( P = [a_{ij}] \) be a \( 3 \times 3 \) matrix. The elements of matrix \( Q \) are defined as:
\[ b_{ij} = 2^{i+j} a_{ij} \]
for \( 1 \leq i, j \leq 3 \).

### Step 2: Write out the matrices
The matrix \( P \) can be represented as:
\[ P = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \]

The matrix \( Q \) can be represented as:
\[ Q = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix} = \begin{bmatrix} 2^{1+1} a_{11} & 2^{1+2} a_{12} & 2^{1+3} a_{13} \\ 2^{2+1} a_{21} & 2^{2+2} a_{22} & 2^{2+3} a_{23} \\ 2^{3+1} a_{31} & 2^{3+2} a_{32} & 2^{3+3} a_{33} \end{bmatrix} \]

This simplifies to:
\[ Q = \begin{bmatrix} 4 a_{11} & 8 a_{12} & 16 a_{13} \\ 8 a_{21} & 16 a_{22} & 32 a_{23} \\ 16 a_{31} & 32 a_{32} & 64 a_{33} \end{bmatrix} \]

### Step 3: Factor out common terms
We can factor out powers of 2 from each row of the matrix \( Q \):
\[ Q = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 16 \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \]
This can be expressed as:
\[ Q = 2^2 \cdot 2^3 \cdot 2^4 \cdot P = 2^9 \cdot P \]

### Step 4: Calculate the determinant of \( Q \)
Using the property of determinants that states \( \text{det}(cA) = c^n \cdot \text{det}(A) \) for an \( n \times n \) matrix \( A \):
\[ \text{det}(Q) = \text{det}(2^9 \cdot P) = 2^{9} \cdot \text{det}(P) \]

Given that \( \text{det}(P) = 2 \):
\[ \text{det}(Q) = 2^{9} \cdot 2 = 2^{10} \]

### Final Answer
Thus, the determinant of the matrix \( Q \) is:
\[ \text{det}(Q) = 2^{10} \]

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