TS EAMCET 10 May 2024 Shift 2 Solved Paper

To evaluate the determinant of the matrix:$\begin{vmatrix} -\frac{bc}{a^2} & \frac{c}{a} & \frac{b}{a} \\ \frac{c}{b} & -\frac{ac}{b^2} & \frac{a}{b} \\ \frac{b}{c} & \frac{a}{c} & -\frac{ab}{c^2} \end{vmatrix}$we can simplify the process by factoring out $\frac{1}{a^2}, \frac{1}{b^2}$, and $\frac{1}{c^2}$ from rows $R_1, R_2$, and $R_3$ respectively. This allows us to focus on a simpler determinant:$\Rightarrow \;\; \frac{1}{a^2 b^2 c^2} \begin{vmatrix} -bc & ac & ab \\ bc & -ac & ab \\ bc & ac & -ab \end{vmatrix}$Next, apply row operations to simplify further: replace $R_2$ with $R_2+R_1$ and $R_3$ with $R_3+R_1$ :$\Rightarrow \;\; \frac{1}{a^2 b^2 c^2} \begin{vmatrix} -bc & ac & ab \\ 0 & 0 & 2ab \\ 0 & 2ac & 0 \end{vmatrix}$This matrix is now triangular, allowing us to compute the determinant by multiplying the diagonal elements:$\Rightarrow \;\; \frac{1}{a^2 b^2 c^2} \times (4 a^2 b^2 c^2) =4$Thus, the value of the determinant is 4 .