Find the Determinant [[1,1,1],[2,-1,3],[4,5,1]]

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)−(3)*(5)=−1−15=−16

3. Calculate the second minor by multiplying the second element 1 by the determinant of the 2×2 matrix remaining after removing the first row and second column, applying a negative sign according to the checkerboard pattern.

−det(2)=−((2)*(1)−(3)*(4))=−(2−12)=10

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)*(5)−(−1)*(4)=10+4=14

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

−16+10+14=8

Final Answer

det(1)=8

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