Problem 7 - Precalculus: Mathematical for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{ll} \frac{1}{2} & \frac{1}{8} \\ 1 & \frac{1}{2} \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is \( \frac{1}{8} \).

Step 1: Understand the Determinant Formula for 2x2 Matrices

To find the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), we use the formula: \( \text{det}(A) = ad - bc \).

Step 2: Identify Matrix Elements

Identify the elements of the matrix: \( a = \frac{1}{2} \), \( b = \frac{1}{8} \), \( c = 1 \), and \( d = \frac{1}{2} \).

Step 3: Apply the Determinant Formula

Substitute the identified elements into the determinant formula: \( \text{det}(A) = \left(\frac{1}{2} \cdot \frac{1}{2}\right) - \left(\frac{1}{8} \cdot 1\right) \).

Step 4: Calculate Each Term Separately

Calculate \( \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \) and \( \frac{1}{8} \cdot 1 = \frac{1}{8} \).

Step 5: Find the Determinant

Subtract the second term from the first: \( \frac{1}{4} - \frac{1}{8} \). Convert \( \frac{1}{4} \) to a common denominator of 8: \( \frac{2}{8} \). So, the calculation becomes \( \frac{2}{8} - \frac{1}{8} = \frac{1}{8} \).

Key Concepts

2x2 Matrices

Matrices are rectangles filled with numbers, arranged in rows and columns. A 2x2 matrix is one of the simplest forms, having two rows and two columns. This makes them easier to manage yet powerful for many computations. They look like this:
\[\begin{bmatrix} a & b \ c & d \end{bmatrix} \]
Here, "a" and "b" are the elements of the first row, while "c" and "d" are for the second row. Finding the determinant of this simple structure can help in multiple areas including solving equations and understanding matrix properties.

• Key feature: It provides a single number (the determinant) that holds important information about the matrix.
• Not all matrices have a determinant, but 2x2 matrices always do unless they contain undefined elements.

Matrix Elements

Matrix elements are the individual numbers placed into the rectangles of a matrix. They are the building blocks of matrix operations.
In a 2x2 matrix, each position has a significant role:

• Top-left is 'a'
• Top-right is 'b'
• Bottom-left is 'c'
• Bottom-right is 'd'

In our exercise, the given matrix:\[\begin{bmatrix} \frac{1}{2} & \frac{1}{8} \ 1 & \frac{1}{2} \end{bmatrix} \]
It has elements: \(a = \frac{1}{2}, b = \frac{1}{8}, c = 1, \) and \(d = \frac{1}{2}\). These elements are crucial for performing calculations like finding the determinant. It's important to note where each element is positioned, as it changes the matrix’s meaning and the outcome of calculations.

Mathematical Calculation

To compute something from a 2x2 matrix, like the determinant, you need a specific formula. This formula for our matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by:
\[\text{det}(A) = ad - bc\]
For our specific elements ("a": \(\frac{1}{2}\), "b": \(\frac{1}{8}\), "c": 1, "d": \(\frac{1}{2}\)), the calculation looks like this:
1. **Multiply 'a' and 'd'**: \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)2. **Multiply 'b' and 'c'**: \(\frac{1}{8} \times 1 = \frac{1}{8}\)3. **Subtract second product from first**: \(\frac{1}{4} - \frac{1}{8}\)To perform the subtraction, convert \(\frac{1}{4}\) to a common denominator:
\(\frac{1}{4} = \frac{2}{8}\)
Then, subtract to get \(\frac{1}{8}\). The answer \(\frac{1}{8}\) is the determinant, revealing whether the matrix operations are valid and the type of transformations it represents.