Final answer:
The determinant of the matrix [[9,2,-1],[5,0,0],[7,1,1]] is calculated using cofactor expansion along the second row and is found to be 0.
Step-by-step explanation:
To find the determinant of the matrix [[9,2,-1],[5,0,0],[7,1,1]], we use the method of cofactor expansion. Since the second row has the simplest elements (including a zero), it's convenient to expand along this row.
- Multiply 5 by the determinant of the 2x2 matrix obtained by removing the second row and first column, which is |0, 0; 1, 1|.
- Since there's a zero in the second row and second column, we can skip multiplying it by its corresponding 2x2 matrix.
- Multiply 0 by the determinant of the 2x2 matrix formed by removing the second row and third column, but since it will be zero, this step can also be skipped.
Now, calculating the determinant:
- Determinant of |0, 0; 1, 1| = (0*1 - 0*1) = 0
Thus:
- (5)(0) - (0)(Determinant of the 2x2 matrix) = 0
Therefore, the determinant of the matrix [[9,2,-1],[5,0,0],[7,1,1]] is 0.