Find the Determinant [[x+1,1,1],[1,x-1,1],[1,1,x]]

Problem

det(x+1)

Solution

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

D=(x+1)*det(x−1)−1*det(1)+1*det(1)

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

det(x−1)=x*(x−1)−1=x2−x−1

det(1)=x−1

det(1)=1−(x−1)=2−x

3. Substitute the results back into the expansion expression.

D=(x+1)*(x2−x−1)−(x−1)+(2−x)

4. Expand the products and combine like terms.

(x+1)*(x2−x−1)=x3−x2−x+x2−x−1=x3−2*x−1

D=x3−2*x−1−x+1+2−x

5. Simplify the final expression by grouping the x terms and constants.

D=x3−4*x+2

Final Answer

det(x+1)=x3−4*x+2

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