First slide

Introduction to Determinants

Question

Let $A = [a_{i j}] a n d B = [b_{i j}]$ be two $3 X 3$ real matrices such that $b_{i j} = {(3)}^{(i + j - 2)} a_{j i},$ where $i, j = 1, 2, 3.$ If the determinant of $B$ is 81 then the determinant of $A$ is

Moderate

A

$\frac{1}{81}$
B

$\frac{1}{3}$
C

$\frac{1}{9}$
D

3

Solution

$\begin{matrix} |\begin{matrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{matrix}| = |\begin{matrix} 3^{0} a_{11} & 3 a_{12} & 9 a_{13} \\ 3 a_{21} & 9 a_{22} & 27 a_{23} \\ 9 a_{31} & 27 a_{32} & 81 a_{33} \end{matrix}| \\ = 81 \times 9 |\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}| \\ = 81 \times 9 \times | A | \\ 81 = 81 \times 9 \times | A | \end{matrix}$
Therefore $|A| = \frac{1}{9}$

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