Find the determinant of the matrix \(\begin{vmatrix} \rm x-a & \rm y-b & \rm z-c\\ \rm a & \rm b & \rm c \\ \rm x & \rm y & \rm z \end{vmatrix}\)

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