Find the Determinant

Problem

det(2)

Solution

1. Perform row operations to create zeros in the first column. Subtract 2 times the second row from the first row ((R_1)→(R_1)−2*(R_2), subtract 2 times the second row from the third row ((R_3)→(R_3)−2*(R_2), and subtract the second row from the fourth row ((R_4)→(R_4)−(R_2).

det(0)

2. Expand along the first column to reduce the matrix to a 3×3 determinant. Since the only non-zero entry in the first column is at (a_21) the determinant is (−1)(2+1)⋅1 times the minor.

−1⋅det(1)

3. Simplify the 3×3 matrix by adding the first row to the second row ((R_2)→(R_2)+(R_1) to create another zero.

−1⋅det(1)

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

−1⋅(1⋅det(−6))

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

det(−6)=(−6)*(2)−(−1)*(−2)

−12−2=−14

6. Multiply by the external factor to find the final value.

−1⋅(−14)=14

Final Answer

det(2)=14

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