Find the determinant of the matrix \(\begin{vmatrix} \rm z & \rm x & \rm y\\ \rm 1 & \rm 1 & \rm 1 \\ \rm x + y & \rm y+z & \rm z+x \end{vmatrix}\)

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:

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

Apply R1 → R1 + R₃

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

Taking common (x + y + z) from Row 1, we get

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

As we can see that the first and the second 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.

∴\(\begin{vmatrix} \rm z & \rm x & \rm y\\ \rm 1 & \rm 1 & \rm 1 \\ \rm x + y & \rm y+z & \rm z+x \end{vmatrix}\) = 0