Find the Determinant

Problem

det(x)

Solution

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

det(x)

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

(ex)*(ex)*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(−x2)

5. Factor out −x2 from the first row of the 2×2 determinant.

e(2*x)*(−1)*(−x2)*det(1)

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

x2*e(2*x)*((1)*(2+4*x+x2)−(1+x)*(2+x))

7. Expand and simplify the expression inside the parentheses.

x2*e(2*x)*(2+4*x+x2−(2+3*x+x2))

x2*e(2*x)*(2+4*x+x2−2−3*x−x2)

x2*e(x)(2*x)

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

x3*e(2*x)

Final Answer

det(x)=x3*e(2*x)

Want more problems? Check here!