Find the Determinant

Problem

det(2)

Solution

1. Identify the best row or column for cofactor expansion. Row 3 is the most efficient choice because it contains three zeros and only one non-zero entry.

2. Apply the cofactor expansion along the third row. The entry is (a_33)=2 and its position (3,3) has a positive sign since (−1)(3+3)=1

2⋅det(2)

3. Expand the resulting 3×3 determinant along its second row, which contains two zeros. The entry is (a_21)=−1 and its position (2,1) has a negative sign since (−1)(2+1)=−1

2⋅(−(−1))⋅det(−1)

4. Simplify the scalar multipliers outside the determinant.

2⋅1⋅det(−1)

2⋅det(−1)

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

det(−1)=(−1)*(−2)−(2)*(3)

2−6=−4

6. Multiply the result by the previous scalar factor to find the final determinant value.

2⋅(−4)=−8

Final Answer

det(2)=−8

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