Find the Determinant

Problem

det(1)

Solution

1. Perform row operations to simplify the matrix. Subtract 2 times the first row from the fourth row ((R_4)→(R_4)−2*(R_1) to create zeros in the first column.

(R_4)→[[2−2*(1),4−2*(2),1−2*(4),6−2*(4)]]=[[0,0,−7,−2]]

2. Continue row operations to eliminate other elements in the first column. Subtract 8 times the first row from the second row ((R_2)→(R_2)−8*(R_1) and 7 times the first row from the third row ((R_3)→(R_3)−7*(R_1).

(R_2)→[[0,−10,−23,−30]]

(R_3)→[[0,−20,−23,−21]]

3. Expand along the first column now that it contains mostly zeros. The determinant of the 4×4 matrix is equal to 1 times the determinant of the remaining 3×3 matrix.

det(A)=1⋅det(−10)

4. Simplify the 3×3 matrix by subtracting 2 times the first row from the second row ((R_2)→(R_2)−2*(R_1) to create a zero in the first column.

(R_2)→[[−20−2*(−10),−23−2*(−23),−21−2*(−30)]]=[[0,23,39]]

5. Expand along the first column of the 3×3 matrix.

det(A)=−10⋅det(23)

6. Calculate the 2×2 determinant using the formula a*d−b*c

det(23)=(23)*(−2)−(39)*(−7)

−46−(−273)=227

7. Multiply by the remaining factor to find the final result.

−10⋅227=−2270

Final Answer

det(1)=−2270

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