Let P = [ a i j ] be a 3 × 3 matrix and let Q = [ b i j ] , where b i j = 2 ? i + j a i j for 1 i , j 3 . If the determinant of P is 2, then the determinant of the matrix Q is [2012]

Question

Let $P = [a_{i j}]$ be a $3 \times 3$ matrix and let $Q = [b_{i j}],$ where $b_{i j} = 2^{i + j} a_{i j}$ for $1 \leq i, j \leq 3$ . If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is [2012]

Smart Achievers · Accepted Answer

(4)

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

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

$= 2^{9} \cdot 2 \cdot 2^{2}$ $| \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} |$

$= 2^{12} \times | P | = 2^{12} \times 2 = 2^{13}$

Solution

Q.
Let $P = [a_{i j}]$ be a $3 \times 3$ matrix and let $Q = [b_{i j}],$ where $b_{i j} = 2^{i + j} a_{i j}$ for $1 \leq i, j \leq 3$ . If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is [2012]

1 $2^{10}$

2 $2^{11}$

3 $2^{12}$

4 $2^{13}$

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Solution

Q. Let P=[aij] be a 3×3 matrix and let Q=[bij], where bij=2 i+jaij for 1≤i, j≤3. If the determinant of P is 2, then the determinant of the matrix Q is [2012]

1 210

2 211

3 212

4 213