Problem 19 - Intermediate Algebra

Question

Evaluate each of the following determinants. $$ \left|\begin{array}{rr} \frac{3}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{2}{5} \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is \(-\frac{7}{20}\).

Step 1: Introduction to the Determinant of a 2x2 Matrix

To find the determinant of a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), use the formula: \(\text{det} = ad - bc\). This exercise involves calculating the determinant of the matrix given by: $$\begin{pmatrix} \frac{3}{2} & -\frac{1}{2} \ \frac{1}{2} & -\frac{2}{5} \end{pmatrix}$$.

Step 2: Identify Matrix Components

Identify and assign the components of the matrix to the variables in the formula. In the given matrix, let \(a = \frac{3}{2}\), \(b = -\frac{1}{2}\), \(c = \frac{1}{2}\), and \(d = -\frac{2}{5}\).

Step 3: Compute \(ad\)

Calculate the product \(ad = \frac{3}{2} \times -\frac{2}{5}\). To do this, multiply the numerators together and the denominators together: \(ad = \frac{3 \times -2}{2 \times 5} = \frac{-6}{10} = -\frac{3}{5}\).

Step 4: Compute \(bc\)

Calculate the product \(bc = -\frac{1}{2} \times \frac{1}{2}\). Again, multiply the numerators and the denominators: \(bc = \frac{-1 \times 1}{2 \times 2} = \frac{-1}{4}\).

Step 5: Subtract the Results

Now, find the determinant by calculating \(ad - bc = -\frac{3}{5} - (-\frac{1}{4})\). This simplifies to: \(ad - bc = -\frac{3}{5} + \frac{1}{4}\).

Step 6: Perform Fraction Subtraction

To subtract these fractions, first convert them to have a common denominator. The least common denominator of 5 and 4 is 20. Convert each fraction: \(-\frac{3}{5} = -\frac{12}{20}\) and \(\frac{1}{4} = \frac{5}{20}\). Subtract: \(-\frac{12}{20} + \frac{5}{20} = -\frac{7}{20}\).

Step 7: Final Result

The determinant of the matrix is \(-\frac{7}{20}\).

Key Concepts

2x2 matrix

A matrix is a rectangular array of numbers arranged in rows and columns. In mathematics, matrices are used to organize data and perform calculations that can solve various kinds of problems. The most basic type of matrix is a 2x2 matrix, which has exactly two rows and two columns.

When you're working with a 2x2 matrix, you typically represent it like this:

• The first row contains two elements, often labeled as \(a\) and \(b\).
• The second row contains two elements, often labeled as \(c\) and \(d\).

The matrix can be written in the form \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), where each letter represents a specific value or number.

One of the important computations you can perform on a 2x2 matrix is finding its determinant. The determinant is a special number that is calculated using the formula \(ad - bc\). This number is essential in understanding several matrix properties, including whether a matrix has an inverse. Understanding how to handle a 2x2 matrix lays the foundation for more complex matrix operations in higher dimensions.

matrix components

When dealing with matrices, it's essential to properly identify and work with each element or component. In a 2x2 matrix, the components are crucial for various calculations, such as determining the determinant.

Let's revisit the matrix we used earlier: \[\begin{pmatrix} \frac{3}{2} & -\frac{1}{2} \ \frac{1}{2} & -\frac{2}{5} \end{pmatrix}\]To calculate the determinant of this 2x2 matrix using the formula \(ad - bc\), you must first label each component:

• \(a = \frac{3}{2}\)
• \(b = -\frac{1}{2}\)
• \(c = \frac{1}{2}\)
• \(d = -\frac{2}{5}\)

By clearly identifying and using these components, you can ensure that your calculations are accurate. Having a methodical approach to labeling and using matrix components reduces errors and leads to better understanding, especially when dealing with more complicated matrices in future studies.

fraction subtraction

Fraction operations are important for many parts of mathematics, including the calculation of determinants in matrices. In our example, the final step to find the determinant involves subtracting two fractions.

Here's a quick overview of the subtraction process:

• Determine a common denominator: The least common denominator of fractions with denominators 5 and 4 is 20, making calculation more manageable.
• Convert: Change \(-\frac{3}{5}\) to \(-\frac{12}{20}\) and \(\frac{1}{4}\) to \(\frac{5}{20}\).
• Subtract: Perform \(-\frac{12}{20} + \frac{5}{20} = -\frac{7}{20}\).

It's essential to master fraction subtraction since it's frequently required in different mathematical problems. By practicing with basic calculations and gradually building up to more complex problems, you'll improve your skills. This will aid you in school assignments and ensure more confidence in your mathematics journey.