Let A be a 3 × 3 matrix with det (A) = 4 . Let R _ i denote the i ^...

Let A = \left[\begin{cases}{a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33}\end{cases}\right]⇒ 2A = \left[\begin{cases}2{a}_{11} & 2{a}_{12} & 2{a}_{13} \\ 2{a}_{21} & 2{a}_{22} & 2{a}_{23} \\ 2{a}_{31} & 2{a}_{32} & 2{a}_{33}\end{cases}\right]

${R}_{2} \to 2{R}_{2} + 5{R}_{3}$

B = \left[\begin{cases}2{a}_{11} & 2{a}_{12} & 2{a}_{13} \\ 4{a}_{21} + 10{a}_{31} & 4{a}_{22} + 10{a}_{32} & 4{a}_{23} + 10{a}_{33} \\ 2{a}_{31} & 2{a}_{32} & 2{a}_{33}\end{cases}\right]

\left|B\right| = \left|$\begin{cases}2{a}_{11} & 2{a}_{12} & 2{a}_{13} \\ 4{a}_{21} + 10{a}_{31} & 4{a}_{22} + 10{a}_{32} & 4{a}_{23} + 10{a}_{33} \\ 2{a}_{31} & 2{a}_{32} & 2{a}_{33}\end{cases}$\right|</p><p>\left|B\right| = \left|\begin{cases}2{a}_{11} & 2{a}_{12} & 2{a}_{13} \\ 4{a}_{21} & 4{a}_{22} & 4{a}_{23} \\ 2{a}_{31} & 2{a}_{32} & 2{a}_{33}\end{cases}\right| + \left|\begin{cases}2{a}_{11} & 2{a}_{12} & 2{a}_{13} \\ 10{a}_{31} & 10{a}_{32} & 10{a}_{33} \\ 2{a}_{31} & 2{a}_{32} & 2{a}_{33}\end{cases}\right|

\left|B\right| = 2 \times 4 \times 2\left|$\begin{cases}{a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33}\end{cases}$\right| + 2 \times 10 \times 2\left|$\begin{cases}{a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{31} & {a}_{32} & {a}_{33} \\ {a}_{31} & {a}_{32} & {a}_{33}\end{cases}$\right|</p><p>\left|B\right| = 16\left|\begin{cases}{a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33}\end{cases}\right| + 0

\left|B\right| = 16\left|A\right|$</p><p>Value of the second determinant is zero. Since Row$2 and Row $3$ are identical.

$| B | = 16 \times 4 = 64\left(∵ | A | = 4\right)$