Find the Determinant [[14,2,14],[-21,2,7],[28,4,-21]]

Problem

det(14)

Solution

1. Factor out common constants from each row to simplify the calculation. Factor 14 from the first row, 7 from the second row, and 7 from the third row.

det(A)=(14)*(7)*(7)*det(1)

2. Factor out 1/7 from the second column to remove fractions.

det(A)=(14)*(7)*(7)*(1/7)*det(1)

3. Simplify the scalar coefficient outside the determinant.

14⋅7⋅7⋅1/7=98

4. Apply cofactor expansion along the first row of the simplified matrix.

det(1)=1*det(2)−1*det(−3)+1*det(−3)

5. Calculate the 2×2 determinants.

det(2)=(2)*(−3)−(1)*(4)=−10

det(−3)=(−3)*(−3)−(1)*(4)=5

det(−3)=(−3)*(4)−(2)*(4)=−20

6. Combine the results of the expansion.

−10−5−20=−35

7. Multiply by the scalar coefficient found in step 3.

98⋅(−35)=−3430

Final Answer

det(14)=−3430

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