Find the Determinant

Problem

det(x)

Solution

1. Simplify the matrix by factoring out e^x from the second and third columns to make the calculations more manageable.

det(x)

2. Factor out the common terms (e^x) from column 2 and (e^x) from column 3 using the properties of determinants.

(e^x)*(e^x)*det(x)

3. Perform row operations to create zeros. Subtract x times Row 2 from Row 1 ((R_1)→(R_1)−x*(R_2).

e^(2*x)*det(0)

4. Expand along the first column since it contains two zeros. The sign for the element in the second row, first column is negative.

e^(2*x)*(−1)*det(−x^2)

5. Factor out −x^2 from the first row of the 2×2 determinant.

e^(2*x)*(−1)*(−x^2)*det(1)

6. Calculate the determinant of the 2×2 matrix using the formula a*d−b*c

x^2*e^(2*x)*((1)*(2+4*x+x^2)−(1+x)*(2+x))

7. Expand and simplify the expression inside the parentheses.

x^2*e^(2*x)*(2+4*x+x^2−(2+3*x+x^2))

x^2*e^(2*x)*(2+4*x+x^2−2−3*x−x^2)

x^2*e(x)^(2*x)

8. Multiply the remaining terms to find the final result.

x^3*e^(2*x)

Final Answer

det(x)=x^3*e^(2*x)

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