Find the Determinant [[1,4,-7],[-1,-3,4],[-2,-4,18]]

Problem

det(1)

Solution

1. Set up the expansion along the first row using the formula for a 3×3 determinant.

2. Calculate the first minor by multiplying the element (a_11)=1 by the determinant of the 2×2 matrix remaining after removing the first row and first column.

1⋅((−3⋅18)−(4⋅−4))=1⋅(−54+16)=−38

3. Calculate the second minor by multiplying the element (a_12)=4 by the determinant of the 2×2 matrix remaining after removing the first row and second column, applying a negative sign.

−4⋅((−1⋅18)−(4⋅−2))=−4⋅(−18+8)=−4⋅(−10)=40

4. Calculate the third minor by multiplying the element (a_13)=−7 by the determinant of the 2×2 matrix remaining after removing the first row and third column.

−7⋅((−1⋅−4)−(−3⋅−2))=−7⋅(4−6)=−7⋅(−2)=14

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

−38+40+14=16

Final Answer

det(1)=16

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