Problem 10 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The determinant of the matrix is \(\frac{2}{3}\).

Step 1: Identify the elements in the matrix

The matrix given in the exercise is \[\begin{bmatrix} \frac{2}{3} & \frac{1}{3} \\ -\frac{1}{2} & \frac{3}{4} \end{bmatrix}\]. From here we can identify the elements as follows: \(a = \frac{2}{3}\), \(b = \frac{1}{3}\), \(c = -\frac{1}{2}\), and \(d = \frac{3}{4}\)

Step 2: Apply the formula for the determinant of a 2x2 matrix

The formula for the determinant of a 2x2 matrix is \(ad - bc\). Substituting the given values gives us \(\frac{2}{3} * \frac{3}{4} - \frac{1}{3} * -\frac{1}{2} = \frac{1}{2} + \frac{1}{6}\)

Step 3: Simplify the result

To simplify the result, add the two fractions: \(\frac{1}{2} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\).

Key Concepts

Understanding the 2x2 Matrix

A 2x2 matrix is a simple, organized arrangement of numbers in two rows and two columns. This forms a square matrix often used in linear algebra. For our specific example, the matrix is written as \[\begin{bmatrix} \frac{2}{3} & \frac{1}{3} \-\frac{1}{2} & \frac{3}{4} \end{bmatrix}\] Each number inside the brackets is called an element. Typically, we label these as:

• a, the top-left element (\(\frac{2}{3}\))
• b, the top-right element (\(\frac{1}{3}\))
• c, the bottom-left element (\(-\frac{1}{2}\))
• d, the bottom-right element (\(\frac{3}{4}\))

A 2x2 matrix is quite manageable and serves as the foundation for understanding larger matrices. Its simplicity allows anyone to practice fundamental concepts of matrix algebra such as determinant calculation.

Mastering Fraction Operations in Matrices

Working with fractions can be a little tricky, especially in matrix operations. Understanding how to handle fractions properly is crucial since matrix elements can often be fractions, as seen in our example. Here’s a brief look at how to manage them:When you multiply fractions, remember:

• Multiply the numerators together.
• Multiply the denominators together.
• For example, \(\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}\)

Adding fractions requires a common denominator:

• For \(\frac{1}{2} + \frac{1}{6}\), convert to common denominator \(\frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\).

Understanding these operations ensures accurate calculations, which is essential when solving for the determinant or any other matrix-related problem.

Calculating Matrix Determinants

Matrix determinants are vital in linear algebra, determining properties of the matrix like invertibility. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is given by the formula:\[det = ad - bc\]This formula is both simple and powerful. Let's break it down using our example:- First, calculate the product \(ad\): - \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\)- Then, calculate the product \(bc\): - \(\frac{1}{3} \times -\frac{1}{2} = -\frac{1}{6}\)- Apply the determinant formula: - \(\frac{1}{2} - \left(-\frac{1}{6}\right) = \frac{1}{2} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\)The result, \(\frac{2}{3}\), is the determinant of the matrix, revealing essential characteristics about its linear transformations. This fundamental concept is instrumental in exploring deeper topics such as eigenvalues and systems of equations.