Problem 4 - Intermediate Algebra

Question

Evaluate each of the following determinants. $$ \left|\begin{array}{ll} 3 & 9 \\ 6 & 4 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -42.

Step 1: Identify the Formula for a 2x2 Determinant

To find the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), use the formula: \( ad - bc \). This formula helps calculate the determinant directly.

Step 2: Substitute the Values into the Formula

Given the matrix \( \begin{pmatrix} 3 & 9 \ 6 & 4 \end{pmatrix} \), identify \(a = 3\), \(b = 9\), \(c = 6\), and \(d = 4\). Substitute these values into the formula for the determinant: \(3 \times 4 - 9 \times 6\).

Step 3: Perform the Multiplications

Calculate each part of the formula: First, \(3 \times 4 = 12\). Then, \(9 \times 6 = 54\).

Step 4: Subtract the Products

Subtract the second product from the first product: \(12 - 54 = -42\). This result is the determinant of the matrix.

Key Concepts

2x2 matrix

A 2x2 matrix is one of the simplest forms of matrices in linear algebra. It consists of two rows and two columns, making it a total of four elements. These elements are usually denoted as \( a \), \( b \), \( c \), and \( d \). The structure looks like this:

• \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \)

2x2 matrices are fundamental because they form the building blocks for larger matrices and more complex matrix operations. They simplify the representation of data and transformations in various fields, such as physics, computer graphics, and statistics.
When dealing with a 2x2 matrix, the determinant is a simple scalar value that can tell us a lot about the matrix, such as whether it is invertible or not.

matrix operations

Matrix operations are procedures that involve adding, subtracting, multiplying, and finding the determinant of matrices. Understanding these operations is crucial for solving many problems in linear algebra. When it comes to a 2x2 matrix, the determinant is calculated using the formula \( ad - bc \). This straightforward operation helps determine properties of the matrix that are essential in linear algebra and other applications.
Other critical matrix operations include matrix addition and matrix multiplication.

• Matrix Addition: Adding matrices involves summing the corresponding elements from each matrix.
• Matrix Multiplication: Involves a more complex process where the elements are multiplied and summed based on their position.
• Matrix Scaling: Elements are multiplied by a constant factor to scale the matrix.

Having a solid understanding of these operations allows you to manipulate matrices effectively in various problem-solving scenarios, particularly when working with systems of equations or transformations.

linear algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, matrices, and systems of linear equations. It provides the tools needed for many fields of science and engineering. Within linear algebra, matrices are used to represent and solve systems of linear equations.
A core concept in linear algebra is the study of determinants, which give vital information about the properties of matrices. For instance:

• If the determinant of a matrix is zero, it suggests that the matrix is singular (non-invertible).
• Determinants help in finding the area or volume in geometry by using matrices.
• Determinants are also useful in the process of finding the inverse of a matrix.

The manipulation and understanding of matrices in linear algebra are crucial for practical applications, such as computer graphics, where transformations are represented using matrices. It also plays a significant role in data science, machine learning, and solving engineering problems.