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Chapter 3: Problem 64

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{rr} x & x \ln x \\ 1 & 1+\ln x \end{array}\right|$$

Short Answer

Expert verified
The determinant of the given matrix is \(x\).

Step by step solution

01

Identify the parts in the determinant

The determinant is represented as: \( \left|\begin{array}{ll} a & b \\ c & d \end{array}\right| \). In this case, \(a = x\), \(b = x \ln x\), \(c = 1\), and \(d = (1 + \ln x)\).
02

Calculate the determinant

The formula to calculate a determinant of a 2x2 matrix is \(ad - bc\). Substitute the values of a, b, c and d in the formula, which gives \(x * (1 + \ln x) - 1 * x \ln x\).
03

Analyze the result

Simplify the expression to finalize the result. The final determinant value is \(x + x \ln x - x \ln x\), which simplifies to \(x\).

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