Problem 11 - Precalculus Mathematics 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 a 2x2 Matrix

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated using the formula \( ad - bc \). In this context, \( a = \frac{1}{2} \), \( b = \frac{1}{8} \), \( c = 1 \), and \( d = \frac{1}{2} \).

Step 2: Substitute Matrix Values into the Determinant Formula

Place the values of the matrix into the determinant formula: \[ \text{det} = \left( \frac{1}{2} \times \frac{1}{2} \right) - \left( \frac{1}{8} \times 1 \right) \]

Step 3: Calculate Each Component of the Determinant Formula

Calculate each part separately: - \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \) - \( \frac{1}{8} \times 1 = \frac{1}{8} \)

Step 4: Solve for the Determinant

Now, subtract the calculated products: \[ \frac{1}{4} - \frac{1}{8} \] To subtract these, convert \( \frac{1}{4} \) to an equivalent fraction with a denominator of 8. Thus, \( \frac{1}{4} = \frac{2}{8} \). Now subtract: \[ \frac{2}{8} - \frac{1}{8} = \frac{1}{8} \]

Step 5: Conclude with the Determinant

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

Key Concepts

Matrix Operations

Matrix operations are fundamental tools in linear algebra and are used to perform calculations on matrices. A matrix is essentially a collection of numbers arranged in rows and columns, like a grid. In the context of this exercise, we're dealing with a 2x2 matrix, meaning two rows and two columns.

Understanding how to manipulate matrices, such as adding, subtracting, or finding products, is crucial in various applications across mathematics, physics, and engineering. The operations on a 2x2 matrix, like the one in our exercise, are generally simpler due to its smaller size, but they build the foundational skills necessary for handling larger matrices.

• **Addition/Subtraction:** Element-wise operations where corresponding elements are added or subtracted.
• **Multiplication:** This involves multiplying matrices in a manner that aligns elements along rows and columns appropriately.
• **Determinant Calculation:** One specific operation that gives us information about the matrix, such as whether it is invertible.

Focusing on determinant calculation, like in this task, helps in understanding more about the matrix structure and its properties. It's not only about finding a number but learning how numbers interact within the grid.

Determinant Calculation

The determinant is a special number that can be calculated from a square matrix. It provides important information about the matrix properties, including whether the matrix is invertible and its potential influence on quadratic equations.

For a 2x2 matrix, the formula for the determinant is straightforward:\[ \text{det} = ad - bc \]where \[a, b, c,\] and\[d\] are the elements of the matrix. This simple formula helps determine the matrix's properties quickly, such as whether it has an inverse.

Using this formula:

• Multiply the top-left and bottom-right elements, \(a\times d\).
• Multiply the top-right and bottom-left elements, \(b\times c\).
• Subtract the second product from the first product to find the determinant.

The result can show if a matrix is singular (determinant is zero) or non-singular (determinant is non-zero). If singular, the matrix does not have an inverse.

Fractions in Determinants

When working with matrices that contain fractions, the process does not change, but the arithmetic can be slightly trickier. In our exercise, each step involves fraction multiplication and subtraction, common in determinant calculations.

Doing arithmetic with fractions requires attention to detail:

• **Multiply Fractions:** The product of two fractions is obtained by multiplying their numerators and denominators. For instance, \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\).
• **Subtract Fractions:** To subtract fractions, convert them to have a common denominator. If you have \(\frac{2}{8} - \frac{1}{8}\), the denominators are already common, making subtraction straightforward.
• **Simplify Results:** Always simplify the result if possible, to make calculations easier and clearer.

Handling fractions correctly ensures accurate calculations in determinant problems. The familiarity with converting and simplifying fractions adds good practice in mathematics.