Problem 1 - Algebra for College Students

Question

For Problems \(1-12\), evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{ll}4 & 3 \\ 2 & 7\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is 22.

Step 1: Understand the Formula for Determinants

For a \(2 \times 2\) matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is calculated using the formula: \(ad - bc\). This means multiplying the \(a\) and \(d\) components together, and the \(b\) and \(c\) components together, and then subtracting the latter from the former.

Step 2: Identify Matrix Components

Given the matrix \(\left|\begin{array}{ll} 4 & 3 \ 2 & 7 \end{array}\right|\), identify the components \(a = 4\), \(b = 3\), \(c = 2\), and \(d = 7\). We will use these values in the determinant formula.

Step 3: Apply the Determinant Formula

Substitute the identified values into the determinant formula: \(ad - bc\). This becomes \(4 \times 7 - 3 \times 2\).

Step 4: Perform the Calculations

Calculate each part of the formula: \(4 \times 7 = 28\) and \(3 \times 2 = 6\). Then subtract these results: \(28 - 6 = 22\).

Step 5: Conclude with the Determinant Value

The evaluated determinant of the matrix \(\left|\begin{array}{ll} 4 & 3 \ 2 & 7 \end{array}\right|\) is \(22\).

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is a simple and foundational concept in linear algebra, representing a grid of numbers arranged in two rows and two columns. Imagine a simple table, where each entry captures relationships or data points. Mathematicians use matrices like these to solve systems of equations, perform transformations, and handle various computations efficiently. This specific type of matrix is essential for higher-level mathematical understanding. Thus, mastering it is crucial to appreciate more complex structures later on.

In a typical \[\begin{pmatrix} a & b \ c & d \end{pmatrix}\]matrix layout:

• The first entry "a" is in the first row, first column.
• "b" is in the first row, second column.
• "c" follows in the second row, first column.
• Finally, "d" is in the second row, second column.

Understanding this layout helps you apply various operations, like finding the determinant in our example.

Identifying the Matrix Components

Before using any mathematical formula on a matrix, it's crucial to identify its components correctly. This step ensures that any computations you perform are based on accurate data, leading to the correct result.

In the given matrix:\[\begin{pmatrix} 4 & 3 \ 2 & 7 \end{pmatrix}\]we can identify the following components:

• "a" is the top-left number, 4.
• "b" is at the top-right position, 3.
• "c" rests in the bottom-left, 2.
• And "d" is at the bottom-right, 7.

This identification mirrors reading a grid from left to right, top to bottom. Each component holds a pivotal role in matrix calculations such as finding determinants.

The Determinant Formula for a 2x2 Matrix

One of the fundamental operations you can perform on a 2x2 matrix is finding its determinant. This particular calculation provides insights into the characteristics and properties of the matrix, such as whether it is invertible.

The formula used to determine the determinant of a 2x2 matrix, \[\begin{pmatrix} a & b \ c & d \end{pmatrix}\], is\[ad - bc\].

To apply the formula:

• Multiply the top-left and bottom-right components, i.e., "a" and "d".
• Multiply the top-right and bottom-left components, "b" and "c".
• Subtract the second product from the first.

In our exercise, substituting the relevant numbers gives:\[4 \times 7 - 3 \times 2\].

Completing these calculations, you derive the determinant value of 22. This value represents a key property of the matrix and tells you that the matrix is invertible because its determinant is not zero.