Here, $$P = {[{a_{ij}}]_{3 \times 3}} = \left[ {\matrix{ {{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr {{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr {{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr } } \right]$$

$$Q = {[{b_{ij}}]_{3 \times 3}} = \left[ {\matrix{ {{b_{11}}} & {{b_{12}}} & {{b_{13}}} \cr {{b_{21}}} & {{b_{22}}} & {{b_{23}}} \cr {{b_{31}}} & {{b_{32}}} & {{b_{33}}} \cr } } \right]$$

where, $${b_{ij}} = {2^{i + j}}{a_{ij}}$$

$$\therefore$$ $$\left| Q \right| = \left| {\matrix{ {4{a_{11}}} & {8{a_{12}}} & {16{a_{13}}} \cr {8{a_{21}}} & {16{a_{22}}} & {32{a_{23}}} \cr {16{a_{31}}} & {32{a_{32}}} & {64{a_{33}}} \cr } } \right|$$

$$ = 4 \times 8 \times 16\left| {\matrix{ {{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr {2{a_{21}}} & {2{a_{22}}} & {2{a_{23}}} \cr {4{a_{31}}} & {4{a_{32}}} & {4{a_{33}}} \cr } } \right|$$

$$ = {2^9} \times 2 \times 4\left| {\matrix{ {{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr {{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr {{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr } } \right|$$

$$ = {2^{12}}.\left| P \right| = {2^{12}}.2 = {2^{13}}$$