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  • Let A be a  matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performing the operation

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 R_2 \rightarrow 2R_2 + 5R_3 on 2A, then det(B) is equal to :
Option: 1 16
Option: 2 80
Option: 3 128
Option: 4 64

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best_answer

\\\text{Let }A=\begin{vmatrix} R_{11} & R_{12} & R_{13} \\ R_{21} & R_{22} & R_{23} \\ R_{31} & R_{32} & R_{33} \end{vmatrix}\\2A=\begin{vmatrix} 2R_{11} & 2R_{12} & 2R_{13} \\ 2R_{21} & 2R_{22} & 2R_{23} \\ 2R_{31} & 2R_{32} & 2R_{33} \end{vmatrix}\\\mathrm{R}_{2} \rightarrow 2 \mathrm{R}_{2}+5 \mathrm{R}_{3}

\mathrm{B}=\begin{vmatrix} 2 \mathrm{R}_{11} & 2 \mathrm{R}_{12} & 2 \mathrm{R}_{13} \\ 4 \mathrm{R}_{21}+10 \mathrm{R}_{31} & 4 \mathrm{R}_{22}+10 \mathrm{R}_{32} & 4 \mathrm{R}_{23}+10 \mathrm{R}_{33} \\ 2 \mathrm{R}_{31} & 2 \mathrm{R}_{32} & 2 \mathrm{R}_{33} \end{vmatrix}

\mathrm{R_2\rightarrow R_2-5R_3}

\mathrm{B}=\begin{vmatrix} 2 \mathrm{R}_{11} & 2 \mathrm{R}_{12} & 2 \mathrm{R}_{13} \\ 4 \mathrm{R}_{21} & 4 \mathrm{R}_{22}& 4 \mathrm{R}_{23} \\ 2 \mathrm{R}_{31} & 2 \mathrm{R}_{32} & 2 \mathrm{R}_{33} \end{vmatrix}

\\|\mathrm{B}|=2 \times 2 \times 4\left|\begin{array}{lll} R_{11} & R_{12} & R_{13} \\ R_{21} & R_{22} & R_{23} \\ R_{31} & R_{32} & R_{33} \end{array}\right| \\ =16 \times 4 \\ =64

As, |A| = 4 (Given)

Posted by

himanshu.meshram

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