Mathematics - Determinant Question with Solution | TestHub

Let $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$$\Rightarrow 2A=\left[\begin{array}{ccc}2a_{11}& 2a_{12}& 2a_{13}\\ 2a_{21}& 2a_{22}& 2a_{23}\\ 2a_{31}& 2a_{32}& 2a_{33}\end{array}\right]$

$R_2\to 2R_2+5R_3$

$B=\left[\begin{array}{ccc}2a_{11}& 2a_{12}& 2a_{13}\\ 4a_{21}+10a_{31}& 4a_{22}+10a_{32}& 4a_{23}+10a_{33}\\ 2a_{31}& 2a_{32}& 2a_{33}\end{array}\right]$

$\left|B\right|=\left|\begin{array}{ccc}2a_{11}& 2a_{12}& 2a_{13}\\ 4a_{21}+10a_{31}& 4a_{22}+10a_{32}& 4a_{23}+10a_{33}\\ 2a_{31}& 2a_{32}& 2a_{33}\end{array}\right|$

$\left|B\right|=\left|\begin{array}{ccc}2a_{11}& 2a_{12}& 2a_{13}\\ 4a_{21}& 4a_{22}& 4a_{23}\\ 2a_{31}& 2a_{32}& 2a_{33}\end{array}\right|+\left|\begin{array}{ccc}2a_{11}& 2a_{12}& 2a_{13}\\ 10a_{31}& 10a_{32}& 10a_{33}\\ 2a_{31}& 2a_{32}& 2a_{33}\end{array}\right|$

$\left|B\right|=2\times 4\times 2 \left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|+2\times 10\times 2 \left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{31}& a_{32}& a_{33}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$

$\left|B\right|=16 \left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right| +0$

$\left|B\right|=16\left|A\right|$

Value of the second determinant is zero. Since Row $2$ and Row $3$ are identical.

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