Find the Determinant [[-5,-2,-1],[3,0,5],[-8,1,-5]]

Problem

det(−5)

Solution

1. Set up the expansion along the first row using the formula for a 3×3 determinant.

2. Calculate the first minor by multiplying −5 by the determinant of the 2×2 matrix remaining after removing the first row and first column.

−5*det(0)=−5*((0)*(−5)−(5)*(1))

−5*(0−5)=25

3. Calculate the second minor by subtracting the product of −2 and the determinant of the 2×2 matrix remaining after removing the first row and second column.

−(−2)*det(3)=2*((3)*(−5)−(5)*(−8))

2*(−15+40)=50

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

−1*det(3)=−1*((3)*(1)−(0)*(−8))

−1*(3−0)=−3

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

25+50−3=72

Final Answer

det(−5)=72

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