Problem 67 - 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. $$\left|\begin{array}{ll} x & \ln x \\ 1 & 1 / x \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant of the given matrix is \(1 - \ln x\).

Step 1: Identify the elements of the matrix

In the provided matrix, the elements can be identified as \(a = x\), \(b = \ln x\), \(c = 1\), and \(d = 1 / x\).

Step 2: Apply the formula for determinants of 2x2 matrices

The determinant of a 2x2 matrix can be found by using the formula ad - bc. Substituting the identified elements into the formula gives \(x * (1 / x) - \ln x * 1\).

Step 3: Simplify the expression

Simplifying the above expression results in 1 - \ln x.