Find the Determinant [[1,1,1],[2,-1,1],[-1,3,-1]]

Problem

det(1)

Solution

1. Set up the expansion along the first row using the cofactor method.

2. Calculate the first minor by multiplying the first element 1 by the determinant of the 2×2 matrix remaining after removing the first row and first column.

det(−1)=(−1)*(−1)−(1)*(3)=1−3=−2

3. Calculate the second minor by multiplying the second element 1 by the negative of the determinant of the 2×2 matrix remaining after removing the first row and second column.

−det(2)=−((2)*(−1)−(1)*(−1))=−(−2+1)=1

4. Calculate the third minor by multiplying the third element 1 by the determinant of the 2×2 matrix remaining after removing the first row and third column.

det(2)=(2)*(3)−(−1)*(−1)=6−1=5

5. Sum the results of the three components to find the total determinant.

1*(−2)+1*(1)+1*(5)=−2+1+5=4

Final Answer

det(1)=4

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