Problem 6 - College Algebra and Calculus: An Applied Approach

Question

Find the determinant of the matrix. $$ \left[\begin{array}{rr} -7 & -4 \\ 8 & 7 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is \(-17\).

Step 1: Identify the values

The given 2x2 matrix is \[\begin{array}{rr} -7 & -4 \ 8 & 7 \end{array}\]. From here, we identify the values a = -7, b = -4, c = 8, and d = 7.

Step 2: Use the formula

To find the determinant of a 2x2 matrix, we substitute the identified values into the determinant formula. Therefore the determinant is \((-7)*7 - (-4)*8\).

Step 3: Simplify the expression

Perform the multiplication and subtraction to simplify the expression. This simplifies to \(-49 - (-32)\) which then simplifies further to \(-49 + 32\). This gives the determinant as \(-17\).

Key Concepts

2x2 matrix

A 2x2 matrix is a simple and commonly used structure in linear algebra. It consists of two rows and two columns of numbers. Each number in the matrix is known as an element. The general form of a 2x2 matrix can be represented as:

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

In this format, \( a, b, c, \) and \( d \) are the elements of the matrix. Working with smaller matrices, like a 2x2, is often the first step in learning about more complex matrix operations and concepts.
They are applied in various fields such as computer graphics, physics, and engineering. Despite its simplicity, the determinant of a 2x2 matrix can provide insight into whether a matrix is invertible or not.

matrix operations

Matrix operations are processes you can perform on matrices, like addition, subtraction, multiplication, and finding the determinant. Each of these operations follows specific rules:

• **Addition and Subtraction:** Performed by adding or subtracting respective elements, only possible if the matrices have the same dimensions.
• **Multiplication:** More complex, involving the rows of the first matrix and the columns of the second, requiring specific conditions for compatibility.
• **Determinant:** A scalar (a single number) that provides important information about the matrix. For a 2x2 matrix, it is calculated with the formula \( ad - bc \).

Understanding these operations is crucial as they form the basis for solving linear equations and understanding transformations in mathematics.
They are key components when studying linear algebra as they help in manipulating and interpreting matrix data efficiently.

linear algebra

Linear algebra is a branch of mathematics that focuses on vectors, vector spaces, and linear transformations. It is the study of lines, planes, and subspaces, but it goes beyond by dealing with multi-dimensional spaces and is crucial for modern mathematical understanding.

Some fundamental concepts in linear algebra include:

• **Vectors and vector spaces:** They define quantities having both magnitude and direction, useful in understanding geometric and physical phenomena.
• **Matrices:** They represent linear transformations and are used extensively in both theoretical and applied mathematics.
• **Determinants and Eigenvalues:** These are properties of matrices that provide insights into matrix behavior and stability.

Linear algebra is foundational in fields such as machine learning, data science, computer graphics, and quantum mechanics.
By gaining a solid grasp of linear algebra, students can better understand the mathematical foundations behind these advanced applications.