Problem 9 - Algebra and Trigonometry

Question

Evaluate each determinant. $$ \left|\begin{array}{cc} {\frac{1}{2}} & {\frac{1}{2}} \\ {\frac{1}{8}} & {-\frac{3}{4}} \end{array}\right| $$

Step-by-Step Solution

Answer

-\frac{7}{16}

Step 1: Identify the elements of the matrix

The elements of the 2x2 matrix are as follows: \(a = \frac{1}{2}\), \(b = \frac{1}{2}\), \(c = \frac{1}{8}\) and \(d = -\frac{3}{4}\).

Step 2: Apply the formula for a 2x2 determinant

The formula for a 2x2 determinant is given by \(ad-bc\). Substitute \(a\), \(b\), \(c\), and \(d\) with their corresponding values and solve.

Step 3: Evaluate the expression

Substituting the values gives: \(\frac{1}{2}*(-\frac{3}{4}) - \frac{1}{2}*\frac{1}{8} = -\frac{3}{8}-\frac{1}{16}\)

Step 4: Simplify the result

Combine the fractions to get the final answer. -\frac{3}{8}-\frac{1}{16} = -\frac{7}{16}

Key Concepts

Understanding the 2x2 Matrix

Matrices are an essential tool in mathematics used to organize numbers in a rectangular format. A 2x2 matrix, as the name suggests, consists of two rows and two columns. This kind of matrix is quite manageable, making it an excellent starting point for understanding more complex matrices. Each number in the matrix is placed in a certain position defined by the row and column it belongs to.

A simple example can look like this:

• First row, first column: This is usually labeled as element "a".
• First row, second column: Labeled as element "b".
• Second row, first column: Labeled as element "c".
• Second row, second column: Labeled as element "d".

This 4-element structure forms the basic matrix we will be working with when calculating determinants.

Exploring Matrix Elements

Within our 2x2 matrix, each element holds a specific value that directly influences any calculations we perform, such as finding a determinant. Understanding these elements can help you solve complex matrix problems with ease.

For the given matrix in the exercise:

• Element "a" is represented by \( \frac{1}{2} \).
• Element "b" is also \( \frac{1}{2} \).
• Element "c" is \( \frac{1}{8} \).
• Element "d" equals \( -\frac{3}{4} \).

Each element must be placed correctly to apply matrix operations. A mistake in identifying these elements can lead to incorrect calculations.

Unpacking the Determinant Formula

The determinant provides a scalar value, offering essential insights into the matrix's properties. For a 2x2 matrix, the determinant formula is straightforward:\[ ext{determinant} = ad - bc\]The elements "a," "b," "c," and "d" are those found in the matrix, and this formula performs operations using these elements to calculate the determinant value.

The resulting number can reveal several things about the matrix, such as whether it is invertible or if it represents a certain geometric transformation. In our example:

• Calculate \(a \times d = \frac{1}{2} imes -\frac{3}{4} = -\frac{3}{8}\).
• Next, calculate \(b \times c = \frac{1}{2} imes \frac{1}{8} = \frac{1}{16}\).
• Subtract the two results: \(-\frac{3}{8} - \frac{1}{16}\).

Working with Fraction Operations

When solving determinants of a 2x2 matrix with fractions, fraction operations become vital. These include multiplication and subtraction, where understanding how to find a common denominator is beneficial.

For multiplication:

• Multiply the numerators together for the new numerator.
• Multiply the denominators together for the new denominator.

In subtraction:

• Find a common denominator between the fractions.
• Convert each fraction to have this common denominator.
• Subtract the numerators while keeping the common denominator.

In our example, the operations result in solving the expression \(-\frac{3}{8} - \frac{1}{16}\).
Finding a common denominator means converting \(-\frac{3}{8}\) to \(-\frac{6}{16}\), allowing you to calculate \(-\frac{6}{16} - \frac{1}{16} = -\frac{7}{16}\). Mastering these operations is essential for working with any matrix that includes fractions.