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माना कि $P=\left[a_1\right]$ एक $3 \times 3$ आव्यूह (matrix) है और $Q=\left[b_1\right]$, जहाँ $b_{\|}=2^{[H]} a_{\|}$जब $1 \leq i, j \leq 3$ है। यदि $P$ के सारणिक (determinant) का मान $2$ है तो आव्यूह $Q$ के सारणिक का मान निम्न है
$2^{10}$
$2^{11}$
$2^{12}$
$2^{13}$
Solution
$\text { Given } P=\left[a_1\right]_{3 \times 3} \quad b_{1 j}=2^{1+1} \text { aij } $
$Q=\left[b_1\right]_{3 \times 3} $
$P=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]|P|=2$
$Q=\left[\begin{array}{lll}b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33}\end{array}\right]=\left[\begin{array}{ccc}4 a_{11} & 8 a_{12} & 16 a_{13} \\ 8 a_{21} & 16 a_{22} & 32 a_{23} \\ 16 a_{31} & 32 a_{32} & 64 a_{33}\end{array}\right]$
Determinant of $Q=\left|\begin{array}{ccc}4 a_{11} & 8 a_{12} & 16 a_{13} \\ 8 a_{21} & 16 a_{22} & 32 a_{23} \\ 16 a_{31} & 32 a_{32} & 64 a_{33}\end{array}\right|$
$=4 \times 8 \times 16\left|\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\ 2 a_{21} & 2 a_{22} & 2 a_{23} \\ 4 a_{31} & 4 a_{32} & 4 a_{33}\end{array}\right|$
$=4 \times 8 \times 16 \times 2 \times 4\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|$
$=2^2 \cdot 2^3 \cdot 2^4 \cdot 2^1 \cdot 2^2 \cdot 2^1 $
$=2^{13}$
