Question

The value of the determinant \(\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix}\) is :

• A. 1
• B. 2
• C. 0
• D. -1

Answer 1

The Correct Option is C

To find the value of the determinant \(\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix}\), let's perform a series of elementary row operations that simplify the process. The determinant can be calculated using any of its rows or columns. Let's simplify the calculations by using row operations:

We know that a determinant's value remains unchanged if we add a scalar multiple of one row to another row.

Step 1: Subtract Row 3 from Row 1.

\(\begin{vmatrix} x+y-1 & y+z-1 & z+x-1 \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix}\)

Step 2: Subtract Row 3 from Row 2.

\(\begin{vmatrix} x+y-1 & y+z-1 & z+x-1 \\ z-1 & x-1 & y-1 \\ 1 & 1 & 1 \end{vmatrix}\)

Now, observe that after these transformations, the first two rows are linearly dependent:

Row 1 + Row 2 = \(\vec{0}\)

This means the determinant is zero because the presence of linearly dependent rows (or columns) causes the determinant of a matrix to be zero.

Thus, the value of the determinant is 0.