Correct Answer - Option 4 : 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:
\(\begin{vmatrix} \rm x-a & \rm y-b & \rm z-c\\ \rm a & \rm b & \rm c \\ \rm x & \rm y & \rm z \end{vmatrix}\)
Apply R3 → R3 - R2
= \(\begin{vmatrix} \rm x-a & \rm y-b & \rm z-c\\ \rm a & \rm b & \rm c \\ \rm x-a & \rm y-b & \rm z-c \end{vmatrix}\)
As we can see that the first and the third row of the given matrix are equal.
We know that, if any two rows (columns) of a matrix are same then the value of the determinant is zero.
\(\begin{vmatrix} \rm x-a & \rm y-b & \rm z-c\\ \rm a & \rm b & \rm c \\ \rm x & \rm y & \rm z \end{vmatrix}\) = 0