Find the Determinant of the Resulting Matrix

Problem

det(3)

Solution

1. Identify the matrix structure and choose the most efficient row or column for expansion. The fourth column contains three zeros, making it the ideal choice for cofactor expansion.

2. Apply the cofactor expansion along the fourth column. The only non-zero entry is (a_44)=−2

3. Determine the sign of the cofactor. Since the entry is at position (4,4) the sign is (−1)(4+4)=1

4. Set up the determinant calculation by multiplying the entry by the determinant of its 3×3 minor.

det(A)=−2⋅det(3)

5. Expand the 3×3 determinant along the third row, which contains a zero.

det((M_44))=−12⋅det(2)+6⋅det(3)

6. Calculate the 2×2 determinants.

det(2)=(2)*(−1)−(−2)*(−6)=−2−12=−14

det(3)=(3)*(−6)−(2)*(5)=−18−10=−28

7. Substitute these values back into the 3×3 expansion.

det((M_44))=−12*(−14)+6*(−28)

det((M_44))=168−168=0

8. Multiply by the remaining factor from the first expansion.

det(A)=−2⋅0=0

Final Answer

det(3)=0

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