Find the determinant of the matrix \(\begin{vmatrix} \rm x-y & \rm y-

1.	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 + z2. x2 + y2 + z23. 04. (x + y + z)2 - xyz
Answer» 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

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 + z2. x2 + y2 + z23. 04. (x + y + z)2 - xyz

Answer» 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