Find the Determinant [[1,-2,1],[5,-15,0],[4,-2,6]]

Problem

det(1)

Solution

1. Set up the expansion along the first row to calculate the determinant of the 3×3 matrix.

2. Calculate the first term by multiplying the element (a_11)=1 by the determinant of its 2×2 minor.

1⋅det(−15)=1*((−15)*(6)−(0)*(−2))

1*(−90−0)=−90

3. Calculate the second term by multiplying the element (a_12)=−2 by the negative of the determinant of its 2×2 minor.

−(−2)⋅det(5)=2*((5)*(6)−(0)*(4))

2*(30−0)=60

4. Calculate the third term by multiplying the element (a_13)=1 by the determinant of its 2×2 minor.

1⋅det(5)=1*((5)*(−2)−(−15)*(4))

1*(−10−(−60))=50

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

−90+60+50=20

Final Answer

det(1)=20

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