Problem 5 - Algebra for College Students

Question

Evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{ll}2 & -3 \\ 8 & -2\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is 20.

Step 1: Understand the Determinant Formula

For a \(2 \times 2\) matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated using the formula: \( ad - bc \). This is an important formula to memorize for evaluating determinants.

Step 2: Identify the Matrix Elements

In the given matrix \( \begin{pmatrix} 2 & -3 \ 8 & -2 \end{pmatrix} \), identify the elements: \( a = 2 \), \( b = -3 \), \( c = 8 \), and \( d = -2 \).

Step 3: Substitute Elements into Determinant Formula

Substitute the identified values into the determinant formula: \((2)(-2) - (-3)(8)\).

Step 4: Perform the Multiplications

Calculate the products: \(2 \times -2 = -4\) and \((-3) \times 8 = -24\).

Step 5: Compute the Determinant

Apply subtraction to the calculated products: \(-4 - (-24) = -4 + 24 = 20\). This is the value of the determinant.

Key Concepts

2x2 Matrix

A \(2 \times 2\) matrix is essentially a way of arranging numbers in two rows and two columns. This type of matrix is fundamental in linear algebra and appears frequently in various mathematical and scientific calculations.
A \(2 \times 2\) matrix takes the form:

• First row: consists of two elements (e.g., 2, -3)
• Second row: also consists of two elements (e.g., 8, -2)

Each number is an element of the matrix, and these elements are crucial for calculating the determinant. Understanding how to identify and manipulate these elements is the first step to performing matrix calculations. This involves simply recognizing the layout of the numbers and knowing which ones belong to which row and column.

Determinant Formula

The determinant of a \(2 \times 2\) matrix is a special number that can be calculated using a specific formula. This formula helps to simplify complex matrix operations and provides insights into certain properties of the matrix, such as whether it is invertible.
The formula to determine the determinant of a \(2 \times 2\) matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is:

• \( ad - bc \)

Here,

• \(a\) and \(d\) are the elements from the main diagonal of the matrix.
• \(b\) and \(c\) are the elements from the off-diagonal of the matrix.

By applying this formula to our matrix, you can calculate its determinant efficiently.

Matrix Calculations

Matrix calculations involve several operations, but one of the most common is finding the determinant. In our example, we derived the elements from the given matrix:

• \(a = 2\)
• \(b = -3\)
• \(c = 8\)
• \(d = -2\)

After identifying these elements, we substitute them into the determinant formula \((ad - bc)\). Carrying out the multiplication operations gives us the results for each term in the formula.
Substitute and multiply:

• Multiply the diagonal elements: \(2 \times -2 = -4\)
• Multiply the off-diagonal elements: \(-3 \times 8 = -24\)

These calculations provide the interim values needed to find the final determinant.

Algebraic Operations

Algebraic operations are integral to solving matrix-related problems. In this case, after determining the products from the multiplication, our next step involves performing the subtraction operation to find the determinant.
From our calculations, we find:

• Main diagonal product: \(-4\)
• Off-diagonal product: \(-24\)

The final algebraic operation is subtraction:

• \(-4 - (-24)\)

Subtracting a negative is equivalent to adding, which simplifies to:

• \(-4 + 24 = 20\)

Thus, using simple algebraic operations allows us to conclude that the determinant of the given matrix is 20, demonstrating the power of such basic arithmetic in solving matrix-oriented problems.