JAMIA 2021 — Mathematics PYQ

Given:

Step 2: Factor out $x,y,\text{ and }z$

Since $x,y,z\ne 0$, we can factor out $x$ from the first row, $y$ from the second row, and $z$ from the third row:

Step 3: Apply row operations

Apply the row operation $R_1\to R_1+R_2+R_3$:

Factor out the common term $(1+\frac{1}{x}+\frac{1}{y}+\frac{1}{z})$ from the first row:

Step 4: Simplify the remaining determinant

Apply column operations $C_2\to C_2-C_1$ and $C_3\to C_3-C_1$:

The determinant of the identity matrix is $1$:

Step 5: Solve for the expression

Since $x,y,z\ne 0$, the product $xyz\ne 0$. Therefore:

Rearranging the terms:

Using negative exponent notation: