Find the Determinant det [[5,3,5],[1,7,8],[9,4,2]]

Problem

det(5)

Solution

1. Apply the expansion formula for a 3×3 determinant along the first row.

2. Calculate the first minor by multiplying the element (a_11)=5 by the determinant of the 2×2 matrix remaining after removing the first row and first column.

5⋅det(7)=5*(7⋅2−8⋅4)

5*(14−32)=5*(−18)=−90

3. Calculate the second minor by multiplying the element (a_12)=3 by the determinant of the 2×2 matrix remaining after removing the first row and second column, applying a negative sign according to the cofactor pattern.

−3⋅det(1)=−3*(1⋅2−8⋅9)

−3*(2−72)=−3*(−70)=210

4. Calculate the third minor by multiplying the element (a_13)=5 by the determinant of the 2×2 matrix remaining after removing the first row and third column.

5⋅det(1)=5*(1⋅4−7⋅9)

5*(4−63)=5*(−59)=−295

5. Sum the results of the three components to find the final determinant value.

−90+210−295=−175

Final Answer

det(5)=−175

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