Find the Determinant

Problem

det(−4/7)

Solution

1. Factor out common denominators from each row to simplify the matrix. We factor out 1/7 from Row 1, 1/14 from Row 2, 1/42 from Row 3, and 1/14 from Row 4.

D=1/7⋅1/14⋅1/42⋅1/14*det(−4)

2. Simplify the scalar product outside the determinant.

1/(7⋅14⋅42⋅14)=1/57624

3. Perform row operations to create zeros in the fourth column. Replace (R_2) with (R_2)−(R_1) (R_3) with (R_3)−(R_1) and (R_4) with (R_4)+(R_1)

det(−4)

4. Expand along the fourth column to reduce the matrix to a 3×3

1⋅(−1)(1+4)*det(−14)

5. Factor out constants from the rows of the 3×3 matrix. Factor −7 from (R_1) 7 from (R_2) and 7 from (R_3)

−1⋅(−7)*(7)*(7)*det(2)

343*det(2)

6. Add (R_2) to (R_3) to create more zeros.

343*det(2)

7. Expand along the second column.

343⋅(−2)⋅(−1)(2+2)*det(2)

−686⋅((2)*(−2)−(1)*(−2))

−686⋅(−4+2)=1372

8. Multiply the result by the scalar factor from step 2.

1372/57624=1/42

Final Answer

det(−4/7)=1/42

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