CUET Preparation Today
Evaluate $\begin{vmatrix} a + x & y & z \\ x & a + y & z \\ x & y & a + z \end{vmatrix}$. |
$a(x + y + z)$ $a^2(x + y + z)$ $a^2(a + x + y + z)$ $a^3(x + y + z)$ |
$a^2(a + x + y + z)$ |
The correct answer is Option (3) → $a^2(a + x + y + z)$ ## We have, $\begin{vmatrix} a + x & y & z \\ x & a + y & z \\ x & y & a + z \end{vmatrix} = \begin{vmatrix} a & -a & 0 \\ 0 & a & -a \\ x & y & a + z \end{vmatrix} \quad \left[ \begin{aligned} &∵R_1 \to R_1 - R_2 \\ &\text{and } R_2 \to R_2 - R_3 \end{aligned} \right]$ $= \begin{vmatrix} a & 0 & 0 \\ 0 & a & -a \\ x & x + y & a + z \end{vmatrix} \quad [∵ C_2 \to C_2 + C_1]$ On expanding along $R_1$, we get $= a(a^2 + az + ax + ay)$ $= a^2 (a + z + x + y)$ |
