Find the Determinant [[-5,0,1],[1,-3,-2],[0,10,6]]

Problem

det(−5)

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 by multiplying each element by its corresponding minor and alternating signs.

det(A)=−5⋅det(−3)−0⋅det(1)+1⋅det(1)

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

det(−3)=(−3)*(6)−(−2)*(10)=−18+20=2

det(1)=(1)*(10)−(−3)*(0)=10−0=10

4. Substitute the values back into the expansion expression.

det(A)=−5*(2)−0+1*(10)

5. Simplify the expression to find the final value.

det(A)=−10+10=0

Final Answer

det(−5)=0

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