Find the Determinant [[1,3,4],[4,5,8],[3,12,14]]

Problem

det(1)

Solution

1. Apply the expansion formula for the determinant along the first row.

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

1⋅((5⋅14)−(8⋅12))

1⋅(70−96)=−26

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

−3⋅((4⋅14)−(8⋅3))

−3⋅(56−24)=−3⋅32=−96

4. Calculate the third minor by multiplying the element 4 by the determinant of the 2×2 matrix remaining after removing the first row and third column.

4⋅((4⋅12)−(5⋅3))

4⋅(48−15)=4⋅33=132

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

−26−96+132=10

Final Answer

det(1)=10

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