Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)(i + j – 2)aji, where i, j = 1, 2, 3. If the determinant of B is 81, then the determinant of A is:

Engineering

Mathematics

Introduction to Matrix

Question

Let A = [a_ij] and B = [b_ij] be two 3 × 3 real matrices such that b_ij = (3)^{(i + j – 2)}a_ji, where i, j = 1, 2, 3. If the determinant of B is 81, then the determinant of A is:

1/9

1/81

1/3

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$| B | = | \begin{matrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{matrix} | = | \begin{matrix} 3^{0} a_{11} & 3^{1} a_{12} & 3^{2} a_{13} \\ 3^{1} a_{21} & 3^{2} a_{22} & 3^{3} a_{23} \\ 3^{2} a_{31} & 3^{3} a_{32} & 3^{4} a_{33} \end{matrix} | \Rightarrow 81 = 3^{3} . 3 {.3}^{2} | A | \Rightarrow 3^{4} = 3^{6} | A | \Rightarrow | A | = \frac{1}{9}$