Find the Determinant

Problem

det(7)

Solution

1. Factor out a constant from the second row to simplify the entries.

det(7)=2*det(7)

2. Perform row operations to create zeros in the second column. Use (R_1)⇒(R_1)+6*(R_2) (R_3)⇒(R_3)+8*(R_2) and (R_4)⇒(R_4)+4*(R_2)

2*det(19)

3. Expand along the second column to reduce the matrix to a 3×3 determinant.

2⋅(−1)⋅(−1)(2+2)*det(19)=−2*det(19)

4. Perform row operations on the 3×3 matrix to further simplify. Use (R_1)⇒(R_1)−(R_3) and (R_2)⇒(R_2)−(R_3)

−2*det(4)

5. Perform row operations again. Use (R_2)⇒(R_2)−2*(R_1) and (R_3)⇒(R_3)−4*(R_1)

−2*det(4)

6. Perform row operations to create a zero in the first column. Use (R_3)⇒(R_3)−(R_2)

−2*det(4)

7. Expand along the first column or continue row operations. Use (R_1)⇒(R_1)+4*(R_2)

−2*det(0)

8. Calculate the determinant of the resulting 2×2 matrix.

−2⋅(−(−1))*det(19)=−2*(19⋅13−87⋅(−9))

9. Simplify the arithmetic.

−2*(247+783)=−2*(1030)=−2060

Final Answer

det(7)=−2060

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