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 :
Solution
$$A = \left[ {\matrix{
{{R_{11}}} & {{R_{12}}} & {{R_{13}}} \cr
{{R_{21}}} & {{R_{22}}} & {{R_{23}}} \cr
{{R_{31}}} & {{R_{32}}} & {{R_{33}}} \cr
} } \right]$$<br><br>$$2A = \left[ {\matrix{
{2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr
{2{R_{21}}} & {2{R_{22}}} & {2{R_{23}}} \cr
{2{R_{31}}} & {2{R_{32}}} & {2{R_{33}}} \cr
} } \right]$$<br><br>${R_2} \to 2{R_2} + 5{R_3}$<br><br>$$B = \left[ {\matrix{
{2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr
{4{R_{21}} + 10{R_{31}}} & {4{R_{22}} + 10{R_{32}}} & {4{R_{23}} + 10{R_{33}}} \cr
{2{R_{31}}} & {2{R_{32}}} & {2{R_{33}}} \cr
} } \right]$$<br><br>${R_2} \to {R_2} - 5{R_3}$<br><br>$$B = \left[ {\matrix{
{2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr
{4{R_{21}}} & {4{R_{22}}} & {4{R_{23}}} \cr
{2{R_{31}}} & {2{R_{32}}} & {2{R_{33}}} \cr
} } \right]$$<br><br>$$\left| B \right| = \left[ {\matrix{
{2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr
{4{R_{21}}} & {4{R_{22}}} & {4{R_{23}}} \cr
{2{R_{31}}} & {2{R_{32}}} & {2{R_{33}}} \cr
} } \right]$$<br><br>$$\left| B \right| = 2 \times 2 \times 4\left| {\matrix{
{{R_{11}}} & {{R_{12}}} & {{R_{13}}} \cr
{{R_{21}}} & {{R_{22}}} & {{R_{23}}} \cr
{{R_{31}}} & {{R_{32}}} & {{R_{33}}} \cr
} } \right|$$<br><br>$= 16 \times 4$<br><br>$= 64$
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
This question is part of PrepWiser's free JEE Main question bank. 274 more solved questions on Matrices and Determinants are available — start with the harder ones if your accuracy is >70%.