Find the Determinant [[1,4,8],[4,7,-4],[8,-4,1]]

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) by the determinant of its 2×2 minor.

1⋅det(7)=1⋅(7⋅1−(−4)⋅(−4))

1⋅(7−16)=−9

3. Calculate the second term by multiplying the element (a_12) by the determinant of its 2×2 minor, applying a negative sign according to the cofactor pattern.

−4⋅det(4)=−4⋅(4⋅1−(−4)⋅8)

−4⋅(4+32)=−4⋅36=−144

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

8⋅det(4)=8⋅(4⋅(−4)−7⋅8)

8⋅(−16−56)=8⋅(−72)=−576

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

−9−144−576=−729

Final Answer

det(1)=−729

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