Find the Determinant [[9,5,6],[2,-3,3],[-9,2,4]]

Problem

det(9)

Solution

1. Set up the expansion along the first row using the formula for a 3×3 determinant.

2. Calculate the first term by multiplying 9 by the determinant of the 2×2 matrix remaining after removing the first row and first column.

9⋅((−3⋅4)−(3⋅2))=9⋅(−12−6)=9⋅(−18)=−162

3. Calculate the second term by multiplying −5 by the determinant of the 2×2 matrix remaining after removing the first row and second column.

−5⋅((2⋅4)−(3⋅−9))=−5⋅(8+27)=−5⋅(35)=−175

4. Calculate the third term by multiplying 6 by the determinant of the 2×2 matrix remaining after removing the first row and third column.

6⋅((2⋅2)−(−3⋅−9))=6⋅(4−27)=6⋅(−23)=−138

5. Sum the results of the three terms to find the final determinant.

−162−175−138=−475

Final Answer

det(9)=−475

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