Find the Determinant [[2,-5,0],[3,-4,12],[3,-2,8]]

Problem

det(2)

Solution

1. Identify the matrix and choose a row or column for expansion. We will expand along the first row because it contains a zero.

2. Apply the cofactor expansion formula for the first row.

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

3. Calculate the first term using the element 2 and its corresponding 2×2 minor.

2⋅det(−4)=2*((−4)*(8)−(12)*(−2))

2*(−32+24)=2*(−8)=−16

4. Calculate the second term using the element −5 Remember to negate this term based on the sign pattern +−+

−(−5)⋅det(3)=5*((3)*(8)−(12)*(3))

5*(24−36)=5*(−12)=−60

5. Calculate the third term using the element 0

0⋅det(3)=0

6. Sum the results of the expansion steps to find the final determinant.

−16−60+0=−76

Final Answer

det(2)=−76

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