The determinant to evaluate is:
111xx+yxyyx+y
Apply row operations to simplify the determinant. Subtract Row 1 from Row 2 and Row 3:
R2→R2−R1
R3→R3−R1
For Row 2:
First element: 1−1=0
Second element: (x+y)−x=y
Third element: y−y=0
For Row 3:
First element: 1−1=0
Second element: x−x=0
Third element: (x+y)−y=x
The determinant becomes:
100xy0y0x
This is an upper triangular matrix with all elements below the main diagonal equal to zero.
For a triangular matrix, the determinant equals the product of the diagonal elements.
The diagonal elements are: 1,y,x
Determinant =1×y×x
Determinant =xy
Therefore, the value of the determinant is xy.