Problem 20 - Intermediate Algebra

Question

Evaluate each of the following determinants. $$ \left|\begin{array}{rr} -\frac{1}{4} & \frac{1}{3} \\ \frac{3}{2} & \frac{2}{3} \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is \(-\frac{2}{3}\).

Step 1: Understanding Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix. For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated using the formula \( ad - bc \).

Step 2: Identifying Matrix Elements

Identify the elements of the matrix: \( a = -\frac{1}{4} \), \( b = \frac{1}{3} \), \( c = \frac{3}{2} \), and \( d = \frac{2}{3} \).

Step 3: Applying the Determinant Formula

Substitute the identified elements into the determinant formula: \( ad - bc = \left(-\frac{1}{4}\right) \times \frac{2}{3} - \frac{1}{3} \times \frac{3}{2} \).

Step 4: Simplifying Calculations

Calculate each product: \(-\frac{1}{4} \times \frac{2}{3} = -\frac{1}{6}\) and \( \frac{1}{3} \times \frac{3}{2} = \frac{1}{2} \).

Step 5: Calculating the Determinant

Perform the subtraction to find the determinant: \(-\frac{1}{6} - \frac{1}{2} = -\frac{1}{6} - \frac{3}{6} = -\frac{4}{6} = -\frac{2}{3}\).

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is a simple square matrix with two rows and two columns. It is represented with elements arranged in a grid-like fashion. For example, a generic 2x2 matrix looks like this: \[\begin{pmatrix}a & b \c & d\end{pmatrix}\] Each letter represents a matrix element, and together they form a collection of numbers that are used to perform various operations, including the calculation of determinants. This type of matrix is foundational in linear algebra and is often used to denote systems with two equations. Recognizing and writing down a 2x2 matrix is a skill that paves the way to their understanding and usage.

Differentiating Matrix Elements

Matrix elements are simply individual numbers or variables located within a matrix. In a 2x2 matrix, you have four elements: \(a, b, c,\) and \(d\). These elements are placed like this: - \(a\) is in the first row, first column - \(b\) is in the first row, second column - \(c\) is in the second row, first column - \(d\) is in the second row, second column Recognizing which number is which helps you apply formulas and perform matrix operations correctly. In the given example, the matrix is: \[\begin{pmatrix}-\frac{1}{4} & \frac{1}{3} \\frac{3}{2} & \frac{2}{3}\end{pmatrix}\] Here, \(a = -\frac{1}{4}\), \(b = \frac{1}{3}\), \(c = \frac{3}{2}\), and \(d = \frac{2}{3}\). Identifying these correctly is critical before applying any formula.

Applying the Determinant Formula

The determinant of a 2x2 matrix is an essential concept used to determine if a matrix is invertible and useful in transformations. To calculate this, you apply the determinant formula. For a 2x2 matrix \[\begin{pmatrix}a & b \c & d\end{pmatrix}\] The formula is: \[ ext{Determinant} = ad - bc \] You multiply the top left and bottom right elements, \(a\) and \(d\), and subtract the product of the top right and bottom left elements, \(b\) and \(c\). For our example matrix, this means substituting in the values: \[ (-\frac{1}{4}) \times \frac{2}{3} - \frac{1}{3} \times \frac{3}{2}\] Each step in the calculation must be done carefully to ensure accuracy.

Finding the Scalar Value of a Determinant

The determinant of a matrix is a scalar value, meaning it's a single number extracted from operations on the matrix elements. This value reveals important properties of the matrix, such as whether it's invertible. For a 2x2 matrix, the determinant is the result from the formula \(ad - bc\). Continuing the example, compute 'ad' and 'bc': - \((-\frac{1}{4}) \times \frac{2}{3} = -\frac{1}{6}\) - \(\frac{1}{3} \times \frac{3}{2} = \frac{1}{2}\) Subtract \(-\frac{1}{6}\) from \(\frac{1}{2}\): \[ -\frac{1}{6} - \frac{1}{2} = -\frac{2}{3} \] This scalar value, \(-\frac{2}{3}\), is the determinant of the matrix and tells us about the matrix's properties and transformations.