Question

Find the determinant of the matrix \(\begin{vmatrix} \rm x-y & \rm y-z & \rm z-x\\ \rm y-z & \rm z-x & \rm x-y \\ \rm z-x & \rm x-y & \rm y-z \end{vmatrix}\)
1. x + y + z
2. x² + y² + z²
3. 0
4. (x + y + z)² - xyz

Answer 1

Correct Answer - Option 3 : 0

Concept:

Properties of Determinant of a Matrix:

• If each entry in any row or column of a determinant is 0, then the value of the determinant is zero.
• For any square matrix say A, |A| = |AT|.
• If we interchange any two rows (columns) of a matrix, the determinant is multiplied by -1.
• If any two rows (columns) of a matrix are same then the value of the determinant is zero.

 

Calculation:

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

Apply R1 → R1 + R₂ + R₃, We get

= \(\begin{vmatrix} \rm 0 & \rm 0 & \rm 0\\ \rm y-z & \rm z-x & \rm x-y \\ \rm z-x & \rm x-y & \rm y-z \end{vmatrix}\)

As we can see that the entry of the first row is zero. 

We know that,

If each entry in any row or column of a determinant is 0, then the value of the determinant is zero.

∴ \(\begin{vmatrix} \rm x-y & \rm y-z & \rm z-x\\ \rm y-z & \rm z-x & \rm x-y \\ \rm z-x & \rm x-y & \rm y-z \end{vmatrix}\) = 0