31.

Let $\mathrm{A} = \left[\begin{array}{cc}\mathrm{x} + \mathrm{y}& \mathrm{y}\\ 2\mathrm{x}& \mathrm{x} - \mathrm{y}\end{array}\right], \mathrm{B} = \left[\begin{array}{c}2\\ - 1\end{array}\right] \mathrm{and} \mathrm{C} = \left[\begin{array}{c}3\\ 2\end{array}\right]$ If AB = C, then what is the value of the determinant of the matrix A ?

•  - 10

•  - 14

•  - 24

•  - 34

$$\begin{gathered} \mathrm{It} \mathrm{is} \mathrm{given} \mathrm{that} \mathrm{AB} = \mathrm{C} \\ \Rightarrow \left[\begin{array}{cc}\mathrm{x} \hspace{0.17em}+ \mathrm{y}& \mathrm{y}\\ 2\mathrm{x}& \mathrm{x} - \mathrm{y}\end{array}\right]\left[\begin{array}{c}2\\ - 1\end{array}\right] = \left[\begin{array}{c}3\\ 2\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}2\mathrm{x} +\hspace{0.17em}2\mathrm{y} - \mathrm{y}\\ 4\mathrm{x} - \mathrm{x} +\hspace{0.17em}\mathrm{y}\end{array}\right] = \left[\begin{array}{c}3\\ 2\end{array}\right] \\ \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{two} \mathrm{matrix} \mathrm{are} \mathrm{equal} \mathrm{when} \mathrm{corresponding} \\ \mathrm{elements} \mathrm{of} \mathrm{both} \mathrm{matrix} \mathrm{are} \mathrm{equal}. \\ \Rightarrow 2\mathrm{x} +\hspace{0.17em}\mathrm{y} = 3 ...\left(\mathrm{i}\right) \\ \Rightarrow 3\mathrm{x} + \mathrm{y} = 2 ...\left(\mathrm{ii}\right) \\ \mathrm{Subtract} \left(1\right) \mathrm{from} \left(2\right) \\ \Rightarrow \mathrm{x} = - 1 \\ \mathrm{Put} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{x} \mathrm{in} \left(1\right) \\ \Rightarrow \mathrm{y} = 5 \\ \mathrm{Hence}, \mathrm{A} = \left[\begin{array}{cc}4& 5\\ - 2& - 6\end{array}\right] \\ \mathrm{Now}, \left|\mathrm{A}\right| = \left|\begin{array}{cc}4& 5\\ - 2& - 6\end{array}\right| \\ = - 24 +\hspace{0.17em}10 = - 14 \end{gathered}$$