JEE MAIN - Mathematics (2021 - 25th February Evening Shift - No. 9)
Let A be a 3 $$\times$$ 3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performing the operation R2 $$ \to $$ 2R2 + 5R3 on 2A, then det(B) is equal to :
64
16
128
80
Explanation
$$A = \left[ {\matrix{
{{R_{11}}} & {{R_{12}}} & {{R_{13}}} \cr
{{R_{21}}} & {{R_{22}}} & {{R_{23}}} \cr
{{R_{31}}} & {{R_{32}}} & {{R_{33}}} \cr
} } \right]$$
$$2A = \left[ {\matrix{ {2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr {2{R_{21}}} & {2{R_{22}}} & {2{R_{23}}} \cr {2{R_{31}}} & {2{R_{32}}} & {2{R_{33}}} \cr } } \right]$$
$${R_2} \to 2{R_2} + 5{R_3}$$
$$B = \left[ {\matrix{ {2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr {4{R_{21}} + 10{R_{31}}} & {4{R_{22}} + 10{R_{32}}} & {4{R_{23}} + 10{R_{33}}} \cr {2{R_{31}}} & {2{R_{32}}} & {2{R_{33}}} \cr } } \right]$$
$${R_2} \to {R_2} - 5{R_3}$$
$$B = \left[ {\matrix{ {2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr {4{R_{21}}} & {4{R_{22}}} & {4{R_{23}}} \cr {2{R_{31}}} & {2{R_{32}}} & {2{R_{33}}} \cr } } \right]$$
$$\left| B \right| = \left[ {\matrix{ {2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr {4{R_{21}}} & {4{R_{22}}} & {4{R_{23}}} \cr {2{R_{31}}} & {2{R_{32}}} & {2{R_{33}}} \cr } } \right]$$
$$\left| B \right| = 2 \times 2 \times 4\left| {\matrix{ {{R_{11}}} & {{R_{12}}} & {{R_{13}}} \cr {{R_{21}}} & {{R_{22}}} & {{R_{23}}} \cr {{R_{31}}} & {{R_{32}}} & {{R_{33}}} \cr } } \right|$$
$$ = 16 \times 4$$
$$ = 64$$
$$2A = \left[ {\matrix{ {2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr {2{R_{21}}} & {2{R_{22}}} & {2{R_{23}}} \cr {2{R_{31}}} & {2{R_{32}}} & {2{R_{33}}} \cr } } \right]$$
$${R_2} \to 2{R_2} + 5{R_3}$$
$$B = \left[ {\matrix{ {2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr {4{R_{21}} + 10{R_{31}}} & {4{R_{22}} + 10{R_{32}}} & {4{R_{23}} + 10{R_{33}}} \cr {2{R_{31}}} & {2{R_{32}}} & {2{R_{33}}} \cr } } \right]$$
$${R_2} \to {R_2} - 5{R_3}$$
$$B = \left[ {\matrix{ {2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr {4{R_{21}}} & {4{R_{22}}} & {4{R_{23}}} \cr {2{R_{31}}} & {2{R_{32}}} & {2{R_{33}}} \cr } } \right]$$
$$\left| B \right| = \left[ {\matrix{ {2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr {4{R_{21}}} & {4{R_{22}}} & {4{R_{23}}} \cr {2{R_{31}}} & {2{R_{32}}} & {2{R_{33}}} \cr } } \right]$$
$$\left| B \right| = 2 \times 2 \times 4\left| {\matrix{ {{R_{11}}} & {{R_{12}}} & {{R_{13}}} \cr {{R_{21}}} & {{R_{22}}} & {{R_{23}}} \cr {{R_{31}}} & {{R_{32}}} & {{R_{33}}} \cr } } \right|$$
$$ = 16 \times 4$$
$$ = 64$$
Comments (0)
