Find the Determinant [[2,-8,0],[5,4,1],[0,3,-2]]

Problem

det(2)

Solution

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

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

det(A)=(a_11)*(C_11)+(a_12)*(C_12)+(a_13)*(C_13)

3. Set up the minors for each element in the first row.

det(A)=2*det(4)−(−8)*det(5)+0*det(5)

4. Calculate the 2×2 determinants using the formula a*d−b*c

det(4)=(4)*(−2)−(1)*(3)=−8−3=−11

det(5)=(5)*(−2)−(1)*(0)=−10−0=−10

5. Substitute the values back into the expansion equation.

det(A)=2*(−11)+8*(−10)+0

6. Simplify the expression to find the final value.

det(A)=−22−80

det(A)=−102

Final Answer

det(2)=−102

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