Let `P=[a]"be a "3xx3` matrix and let `Q=[b]=2^(i+j)a_(i) "for" 1 lei, 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)`

Answer

To solve the problem, we need to find the determinant of the matrix \( Q \) given the matrix \( P \) and its determinant. Let's go through the steps systematically.

### Step 1: Define the matrices
We have a matrix \( P \) which is a \( 3 \times 3 \) matrix with a known determinant:
\[ \text{det}(P) = 2 \]
The matrix \( Q \) is defined as:
\[ Q = [b] = 2^{(i+j)} a_{ij} \quad \text{for } 1 \leq i, j \leq 3 \]

### Step 2: Construct the matrix \( Q \)
We will calculate the elements of \( Q \) based on the formula \( Q_{ij} = 2^{(i+j)} a_{ij} \).

- For \( i = 1 \):
- \( j = 1 \): \( Q_{11} = 2^{(1+1)} a_{11} = 2^2 a_{11} \)
- \( j = 2 \): \( Q_{12} = 2^{(1+2)} a_{12} = 2^3 a_{12} \)
- \( j = 3 \): \( Q_{13} = 2^{(1+3)} a_{13} = 2^4 a_{13} \)

- For \( i = 2 \):
- \( j = 1 \): \( Q_{21} = 2^{(2+1)} a_{21} = 2^3 a_{21} \)
- \( j = 2 \): \( Q_{22} = 2^{(2+2)} a_{22} = 2^4 a_{22} \)
- \( j = 3 \): \( Q_{23} = 2^{(2+3)} a_{23} = 2^5 a_{23} \)

- For \( i = 3 \):
- \( j = 1 \): \( Q_{31} = 2^{(3+1)} a_{31} = 2^4 a_{31} \)
- \( j = 2 \): \( Q_{32} = 2^{(3+2)} a_{32} = 2^5 a_{32} \)
- \( j = 3 \): \( Q_{33} = 2^{(3+3)} a_{33} = 2^6 a_{33} \)

Thus, the matrix \( Q \) can be written as:
\[ Q = \begin{bmatrix} 2^2 a_{11} & 2^3 a_{12} & 2^4 a_{13} \\ 2^3 a_{21} & 2^4 a_{22} & 2^5 a_{23} \\ 2^4 a_{31} & 2^5 a_{32} & 2^6 a_{33} \end{bmatrix} \]

### Step 3: Factor out common terms
We can factor out the powers of 2 from each row:
- From the first row, factor out \( 2^2 \)
- From the second row, factor out \( 2^3 \)
- From the third row, factor out \( 2^4 \)

This gives us:
\[ Q = 2^2 \begin{bmatrix} a_{11} & 2 a_{12} & 2^2 a_{13} \\ 2^1 a_{21} & 2^2 a_{22} & 2^3 a_{23} \\ 2^2 a_{31} & 2^3 a_{32} & 2^4 a_{33} \end{bmatrix} \]

### Step 4: Calculate the determinant
The determinant of \( Q \) can be calculated as follows:
\[ \text{det}(Q) = 2^{2 + 3 + 4} \cdot \text{det}\left(\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\right) \]
This simplifies to:
\[ \text{det}(Q) = 2^9 \cdot \text{det}(P) \]

Since we know \( \text{det}(P) = 2 \):
\[ \text{det}(Q) = 2^9 \cdot 2 = 2^{10} \]

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