Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} e^{2 x} & e^{3 x} \\ 2 e^{2 x} & 3 e^{3 x} \end{array}\right|$$

Step-by-Step Solution

Answer

The evaluated determinant is \(e^{5x}\).

Step 1: Identify entries

First identify the entries in the determinant. You have \(a = e^{2x}\), \(b = e^{3x}\), \(c = 2e^{2x}\) and \(d = 3e^{3x}\).

Step 2: Apply determinant formula

Now apply the formula for a 2x2 determinant \(ad - bc\). Plug in the identified entries, so you have: \(ad - bc = (e^{2x})*(3e^{3x}) - (e^{3x})*(2e^{2x})\).

Step 3: Simplify result

Next, simplify the expression. You get \(3e^{5x} - 2e^{5x} = e^{5x}\).