Problem 7 - A Graphical Approach to College Algebra

Question

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}3 & 4 \\\5 & -2\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is -26.

Step 1: Identify the Formula for the Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by the formula \( ad - bc \). Here, the matrix elements are \( a = 3 \), \( b = 4 \), \( c = 5 \), and \( d = -2 \).

Step 2: Substitute the Values into the Formula

Substitute the values of \( a \), \( b \), \( c \), and \( d \) into the determinant formula: \[\text{det} = (3)(-2) - (4)(5)\]

Step 3: Calculate the Products

First, calculate the product \( (3)(-2) = -6 \). Then, calculate the product \( (4)(5) = 20 \).

Step 4: Subtraction to Find the Determinant

Subtract the second product from the first to find the determinant:\[-6 - 20 = -26\]

Step 5: Conclusion

The determinant of the matrix \( \begin{bmatrix} 3 & 4 \ 5 & -2 \end{bmatrix} \) is \(-26\).

Key Concepts

matrix determinant formula

A matrix determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, the calculation process is straightforward and involves using a simple formula. This formula is fundamental in linear algebra and plays a crucial role in solving systems of linear equations.

To determine the determinant of a 2x2 matrix, we take the elements of the matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]. The matrix determinant formula for a 2x2 matrix is\[ \operatorname{det}(A) = ad - bc\]. Here, the letters \(a\), \(b\), \(c\), and \(d\) represent the elements of the matrix, read from left to right and top to bottom. The formula involves multiplying the elements diagonally and subtracting the product of the other diagonal.

Understanding this formula allows you to find the determinant without any complicated computation or technology.

2x2 matrix

A 2x2 matrix is one of the simplest forms of matrices used extensively in linear algebra. It consists of two rows and two columns, creating a square matrix. Each matrix includes four components, written in the following form:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]. The structure of a 2x2 matrix makes it easier to understand the fundamental concepts of matrix operations.

In practical terms, a 2x2 matrix can represent a variety of mathematical concepts and real-world applications.

• Primarily, it is used in linear transformations, where it can represent simple geometrical operations like scaling, rotation, reflection, etc.
• They are also used in economic models, computer graphics, physics, and more.

These matrices provide a foundational basis that can be expanded to understand larger and more complex matrices.

linear algebra

Linear Algebra is a branch of mathematics concerning linear equations, linear functions, and their representations through vectors and matrices. It forms the foundation for many advanced mathematical concepts and applications.

One of the key components of linear algebra is understanding matrices and determinants, such as the 2x2 matrix. They play a significant role in areas such as:

• Solving systems of linear equations: Determinants can be used to determine whether a system of equations has one solution, no solution, or infinite solutions.
• Eigenvalues and eigenvectors: Which are crucial in many fields like physics, engineering, and computer science.
• Linear transformations: Which help in understanding how data points can be manipulated mathematically.

Linear algebra is immensely useful in disciplines beyond mathematics, influencing fields such as data science, machine learning, computer graphics, and more. Understanding the concepts in linear algebra, like matrix determinants, can open the door to a deeper comprehension of how these areas operate.