Find the Determinant [[1,x,-3],[2,-1,2x],[-1,2,-1]]

Problem

det(1)

Solution

1. Apply the expansion formula along the first row to set up the determinant calculation.

1⋅det(−1)−x⋅det(2)+(−3)⋅det(2)

2. Calculate the 2×2 determinants using the formula a*d−b*c for each minor.

1*((−1)*(−1)−(2*x)*(2))−x*((2)*(−1)−(2*x)*(−1))−3*((2)*(2)−(−1)*(−1))

3. Simplify the expressions inside the parentheses.

1*(1−4*x)−x*(−2+2*x)−3*(4−1)

4. Distribute the terms outside the parentheses to expand the polynomial.

1−4*x+2*x−2*x2−9

5. Combine like terms to find the final simplified expression for the determinant.

−2*x2−2*x−8

Final Answer

det(1)=−2*x2−2*x−8

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