Find the Determinant [[9,-5,2],[1,1,3],[0,4,-2]]

Problem

det(9)

Solution

1. Choose a row or column for expansion. We will expand along the first column because it contains a zero, which simplifies the calculation.

2. Apply the cofactor expansion formula along the first column. The determinant is calculated as the sum of each element in the column multiplied by its corresponding cofactor.

D=9⋅det(1)−1⋅det(−5)+0⋅det(−5)

3. Calculate the 2×2 determinants using the formula det(a)=a*d−b*c

det(1)=(1)*(−2)−(3)*(4)=−2−12=−14

det(−5)=(−5)*(−2)−(2)*(4)=10−8=2

4. Substitute the values back into the expansion expression.

D=9*(−14)−1*(2)+0

5. Simplify the expression to find the final value.

D=−126−2

D=−128

Final Answer

det(9)=−128

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