Find the Determinant [[1-x,1,-2],[-1,2-x,1],[0,1,-1-x]]

Problem

det(1−x)

Solution

1. Set up the determinant expansion using the first column to take advantage of the zero entry.

2. Expand along the first column using the formula for a 3×3 determinant.

D=(1−x)*det(2−x)−(−1)*det(1)+0

3. Calculate the 2×2 determinant for the first term.

(2−x)*(−1−x)−(1)*(1)=(−2−2*x+x+x2)−1

x2−x−3

4. Calculate the 2×2 determinant for the second term.

(1)*(−1−x)−(1)*(−2)=−1−x+2

1−x

5. Combine the results into the full expression.

D=(1−x)*(x2−x−3)+1*(1−x)

6. Factor out the common term (1−x) to simplify the polynomial.

D=(1−x)*(x2−x−3+1)

D=(1−x)*(x2−x−2)

7. Expand the product to find the final polynomial form.

D=x2−x−2−x3+x2+2*x

−x3+2*x2+x−2

Final Answer

det(1−x)=−x3+2*x2+x−2

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