Evaluate each determinant in Exercises 1–10.$\(\begin{vmatrix}\[\frac{1}{2}\) & \(\frac{1}{2}\) \(\frac{1}{8}\) & - \(\frac{3}{4}\]\end{vmatrix}\)$

Verified step by step guidance

1. Identify the determinant of the 2x2 matrix given as: \(\begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{8} & -\frac{3}{4} \end{pmatrix}\).
2. Recall the formula for the determinant of a 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is \(ad - bc\).
3. Assign the values: \(a = \frac{1}{2}\), \(b = \frac{1}{2}\), \(c = \frac{1}{8}\), and \(d = -\frac{3}{4}\).
4. Calculate the product \(ad = \left(\frac{1}{2}\right) \times \left(-\frac{3}{4}\right)\) and the product \(bc = \left(\frac{1}{2}\right) \times \left(\frac{1}{8}\right)\).
5. Subtract the two products to find the determinant: \(ad - bc\).

Key Concepts

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This scalar value helps determine properties like invertibility of the matrix and is essential in solving systems of linear equations.

Fraction Arithmetic

When working with fractions in matrices, it is important to correctly perform multiplication and subtraction. Multiplying fractions involves multiplying numerators and denominators, while subtraction requires a common denominator.

Simplification of Expressions

After calculating the determinant expression, simplifying the resulting fraction or mixed number is necessary. This involves reducing fractions to their simplest form to provide a clear and final answer.