Find the Determinant

Problem

det(e^x)

Solution

1. Factor out common terms from each column to simplify the matrix. Factor e^x from the first column, e^(2*x) from the second column, and e^(3*x) from the third column.

e^x⋅e^(2*x)⋅e^(3*x)*det(1)

2. Simplify the exponential product outside the determinant using the rule e^a⋅e^b⋅e^c=e^(a+b+c)

e^(6*x)*det(1)

3. Perform row operations to create zeros in the first column. Subtract the first row from the second row ((R_2)−(R_1)⇒(R_2) and subtract the first row from the third row ((R_3)−(R_1)⇒(R_3).

e^(6*x)*det(1)

4. Expand along the first column to reduce the determinant to a 2×2 matrix.

e^(6*x)*(1⋅det(1))

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

e^(6*x)*(1⋅8−2⋅3)

6. Simplify the resulting expression.

e^(6*x)*(8−6)=2*e^(6*x)

Final Answer

det(e^x)=2*e^(6*x)

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