Find the Determinant

Problem

2⋅det(1)

Solution

1. Factor out common terms from the second column to simplify calculations. Notice that all elements in the second column are multiples of 5.

(C_2)=5⋅([68],[51],[87],[45])

2⋅5⋅det(1)

2. Perform row operations to create zeros in the first column. Subtract the first row from the second and fourth rows, and subtract twice the first row from the third row.

(R_2)→(R_2)−(R_1)

(R_3)→(R_3)−2*(R_1)

(R_4)→(R_4)−(R_1)

10⋅det(1)

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

10⋅1⋅det(−17)

4. Perform row operations on the 3×3 matrix to create more zeros. Subtract twice the third row from the first row.

(R_1)→(R_1)−2*(R_3)

10⋅det(29)

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

10⋅(−1)⋅det(29)

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

det(29)=(29)*(−3)−(3)*(−49)

−87+147=60

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

−10⋅60=−600

Final Answer

2⋅det(1)=−600

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