Problem 7 - Algebra for College Students

Question

Evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{ll}-2 & -3 \\ -1 & -4\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is 5.

Step 1: Recall the Formula for a 2x2 Determinant

Definition 11.1 for finding the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is given by the formula: \( ad - bc \).

Step 2: Identify the Elements of the Matrix

For the given matrix \( \begin{pmatrix} -2 & -3 \ -1 & -4 \end{pmatrix} \), assign the values as follows: \( a = -2 \), \( b = -3 \), \( c = -1 \), and \( d = -4 \).

Step 3: Substitute the Values into the Formula

Substitute the identified values into the determinant formula: \( ad - bc = (-2)(-4) - (-3)(-1) \).

Step 4: Compute the Products

Calculate each product separately: \((-2)(-4) = 8\) and \((-3)(-1) = 3\).

Step 5: Calculate the Determinant

Substitute the product values into the expression: \( 8 - 3 = 5 \). Thus, the determinant is \( 5 \).

Key Concepts

2x2 matrix

In the world of linear algebra, matrices play a vital role, and among them, the 2x2 matrix is one of the simplest forms. A 2x2 matrix is a rectangular array of numbers arranged in two rows and two columns. It is usually written in the format:\[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \]The letters \(a\), \(b\), \(c\), and \(d\) represent the individual elements of the matrix. These elements can be numbers such as integers, fractions, or even more complex numbers or expressions. The organization of these elements forms the structure of the matrix.
Understanding how to work with a 2x2 matrix is fundamental because it lays the foundation for more complex concepts in matrix algebra.

determinant formula

The determinant of a matrix is a special number that can be calculated from its elements, and for a 2x2 matrix, it is particularly straightforward. The determinant is helpful in many mathematical contexts, such as solving systems of linear equations, finding inverses of matrices, and calculating areas in coordinate geometry.
To find the determinant of a 2x2 matrix:

• Consider a matrix with elements \(a\), \(b\), \(c\), and \(d\): \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \)
• The formula to calculate its determinant is \( ad - bc \).

This formula means you multiply the diagonals from top-left to bottom-right (\(a\) times \(d\)) and subtract from it the product of the diagonals from top-right to bottom-left (\(b\) times \(c\)). This simple arithmetic operation allows us to easily determine the determinant of a 2x2 matrix.

matrix operations

Matrix operations are the procedures we perform on matrices. They include operations like addition, subtraction, multiplication, and scalar multiplication. Specifically, when evaluating determinants, focus on steps that involve matrix manipulation and arithmetic calculations.One essential operation in regards to determinants is the multiplication of elements and the application of arithmetic operations. For example, when using the determinant formula for a 2x2 matrix, you perform multiplication across diagonals:

• First, multiply the top-left and bottom-right elements: \( ad \).
• Second, multiply the top-right and bottom-left elements: \( bc \).
• Finally, subtract the second product from the first: \( ad - bc \).

Matrix operations become more intuitive with practice, and they form the backbone for many applied concepts in linear algebra and beyond.