Find the Determinant [[1,2,3],[5,7,4],[2,1,3]]

Problem

det(1)

Solution

1. Set up the expansion along the first row using the cofactor method.

2. Calculate the first term by multiplying the element 1 by the determinant of its 2×2 minor.

1⋅det(7)=1⋅(7⋅3−4⋅1)=17

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

−2⋅det(5)=−2⋅(5⋅3−4⋅2)=−2⋅(7)=−14

4. Calculate the third term by multiplying the element 3 by the determinant of its 2×2 minor.

3⋅det(5)=3⋅(5⋅1−7⋅2)=3⋅(−9)=−27

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

17−14−27=−24

Final Answer

det(1)=−24

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