"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:"

Question

Accepted Answer

"|B| = $$\left| {\matrix{ {{b_{11}}} & {{b_{12}}} & {{b_{13}}} \cr {{b_{21}}} & {{b_{22}}} & {{b_{23}}} \cr {{b_{31}}} & {{b_{32}}} & {{b_{33}}} \cr } } \right|$$

= $$\left| {\matrix{ {{3^0}{a_{11}}} & {{3^1}{a_{12}}} & {{3^2}{a_{13}}} \cr {{3^1}{a_{21}}} & {{3^2}{a_{22}}} & {{3^3}{a_{23}}} \cr {{3^2}{a_{31}}} & {{3^3}{a_{32}}} & {{3^4}{a_{33}}} \cr } } \right|$$

= $${3.3^2}\left| {\matrix{ {{a_{11}}} & {{3^1}{a_{12}}} & {{3^2}{a_{13}}} \cr {{a_{21}}} & {{3^1}{a_{22}}} & {{3^2}{a_{23}}} \cr {{a_{31}}} & {{3^1}{a_{32}}} & {{3^2}{a_{33}}} \cr } } \right|$$

= $${3.3^2}{.3.3^2}\left| {\matrix{ {{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr {{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr {{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr } } \right|$$

= 3⁶.|A|

$\therefore$ 3⁶.|A| = 81

$\Rightarrow$ |A| = ${1 \over 9}$"

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:

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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:

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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: