Question

If $\left|\begin{array}{ccc}{x}^2+x& x+1& x-2\\ 2x^2+3x-1& 3x& 3x-3\\ x^2+2x+3& 2x-1& 2x-1\end{array}\right|=Ax-12$, then the value of $A$ is

• A. $12$
• B. $24$
• C. $-12$
• D. $-24$

Answer 1

The correct option is B $24$

Applying $R_3\to R_3{-R}_{1}R{\to }_2{R}_2{-2R}_1$

$\left|\begin{array}{ccc}{x}^2+x& x+1& x-2\\ x-1& x-2& x+1\\ x+3& x-2& x+1\end{array}\right|$

Applying $R_3\to R_3-R_2$

${}^{}\left|\begin{array}{ccc}{x}^2+x& x+1& x-2\\ x-1& x-2& x+1\\ 4& 0& 0\end{array}\right|$

$=4\left[\right(x+{1)}^2-(x-{2)}^2]$

$=4[2x+4x+1-4]$

$=24x-12$

$Ax-12=24x-12$

$A=24$