Problem 17 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The determinant is 5.

Step 1: Understand the Determinant Formula

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated using the formula \( ad - bc \). For our matrix: \( a = \frac{1}{4}, b = -2, c = \frac{3}{2}, d = 8 \).

Step 2: Multiply Diagonal Elements

Calculate the product of the diagonal elements \( a \) and \( d \): \( \frac{1}{4} \times 8 = 2 \).

Step 3: Multiply Other Diagonal Elements

Calculate the product of the other diagonal elements \( b \) and \( c \): \( -2 \times \frac{3}{2} = -3 \).

Step 4: Subtract Products

Subtract the product from Step 3 from the product in Step 2: \( 2 - (-3) = 2 + 3 = 5 \).

Key Concepts

Matrix Operations

Matrix operations are fundamental in mathematics and are used to solve systems of equations, among other applications. A matrix is essentially a grid of numbers, called elements, arranged into rows and columns. Matrices can store and manage data, perform transformations, and more. Different operations can be performed on matrices, such as addition, subtraction, multiplication, and finding determinants.

• **Addition and Subtraction**: Only matrices of the same size can be added or subtracted. This involves adding or subtracting corresponding elements in the matrices.
• **Multiplication**: Matrix multiplication involves multiplying elements and summing results, but it is not as straightforward as multiplying individual elements directly.
• **Finding Determinants**: This operation helps determine if a matrix has an inverse, and it's crucial for solving linear equations using matrices. It involves specific calculations depending on the matrix size.

Matrix operations employ rules that must be followed precisely to ensure accurate results. Understanding these operations is crucial, especially when dealing with different types of matrices like 2x2 matrices.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices, consisting of two rows and two columns. It is usually represented as:\[\begin{pmatrix}a & b \c & d\end{pmatrix}\]In this setup, each letter, \(a\), \(b\), \(c\), and \(d\), represents an element of the matrix. The position of these elements is crucial for any operations performed on the matrix.

• **Ease of Use**: The simplicity of 2x2 matrices makes them convenient for learning and applying basic concepts of matrices without overwhelming complexity.
• **Applications**: These matrices are utilized in various fields such as physics, computer science, and engineering, primarily due to their manageable size yet powerful mathematical utility.

When dealing with 2x2 matrices, understanding each element's role is essential, especially in operations like finding a determinant. Although small, a 2x2 matrix serves as a foundational tool for learning about larger matrices and more intricate matrix operations.

Determinant Formula

The determinant of a matrix provides vital information about the matrix itself. For a 2x2 matrix, the determinant is a single number calculated using the formula for its specific matrix structure. For a 2x2 matrix represented as:\[\begin{pmatrix}a & b \c & d\end{pmatrix}\]the determinant is calculated with the formula:\[det = ad - bc\]Here, you multiply the elements of the main diagonal, \(a\) and \(d\), and subtract the product of the other diagonal, \(b\) and \(c\).

• **Significance**: The value of the determinant tells us whether the matrix is invertible (non-zero) or singular (zero).
• **Application**: Determinants are used in various calculations, including finding inverse matrices and solutions to systems of linear equations.

In practice, calculating the determinant of a 2x2 matrix is straightforward yet crucial in understanding more complex matrix behavior. This serves as a stepping stone towards tackling larger matrices and advanced linear algebraic concepts.