The value of the determinant needs to be found where x+y+z=0.
Adding all elements in Row 1:
3x+(−x+y)+(−x+z)
=3x−x+y−x+z
=x+y+z
=0
Adding all elements in Row 2:
(x−y)+3y+(z−y)
=x−y+3y+z−y
=x+y+z
=0
Adding all elements in Row 3:
(x−z)+(y−z)+3z
=x−z+y−z+3z
=x+y+z
=0
Each row sums to zero, which means:
C1+C2+C3=000
This shows that one column can be written as a linear combination of the other two, making the columns linearly dependent.
When columns of a matrix are linearly dependent, the determinant equals zero.
The value of the determinant is 0