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

Problem

det(−1)

Solution

1. Apply the expansion formula for the determinant along the first row.

2. Calculate the first minor by multiplying the element (a_11)=−1 by the determinant of the 2×2 matrix remaining after removing the first row and first column.

(−1)*det(3)=(−1)*((3)*(−4)−(−5)*(−3))

(−1)*(−12−15)=27

3. Calculate the second minor by multiplying the element (a_12)=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 cofactor pattern.

−(1)*det(4)=−(1)*((4)*(−4)−(−5)*(−1))

−(1)*(−16−5)=21

4. Calculate the third minor by multiplying the element (a_13)=1 by the determinant of the 2×2 matrix remaining after removing the first row and third column.

(1)*det(4)=(1)*((4)*(−3)−(3)*(−1))

(1)*(−12+3)=−9

5. Sum the results of the three components to find the final determinant value.

27+21−9=39

Final Answer

det(−1)=39

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