Find the Determinant

Problem

det(x)

Solution

1. Apply cofactor expansion along the first row to reduce the 4×4 determinant into 3×3 determinants.

D=x*det(x)−1*det(1)

2. Evaluate the first 3×3 determinant using the rule of Sarrus or cofactor expansion.

det(x)=x*(x2−1)−1*(x−1)+1*(1−x)

=x3−x−x+1+1−x

=x3−3*x+2

3. Evaluate the second 3×3 determinant by expanding along the first column.

det(1)=1*(x2−1)

=x2−1

4. Substitute the results back into the expansion formula from step 1.

D=x*(x3−3*x+2)−1*(x2−1)

5. Distribute and simplify the polynomial expression.

D=x4−3*x2+2*x−x2+1

D=x4−4*x2+2*x+1

Final Answer

det(x)=x4−4*x2+2*x+1

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