Problem 4 - Algebra for College Students

Question

Evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{rr}5 & 3 \\ 6 & -1\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is -23.

Step 1: Set Up the Determinant Formula

For a given \(2 \times 2\) matrix \(\left|\begin{array}{cc} a & b \ c & d \end{array}\right|\), the determinant is found using the formula: \(ad - bc\).

Step 2: Identify Elements of the Matrix

Identify the elements of the given matrix: \(a = 5\), \(b = 3\), \(c = 6\), and \(d = -1\).

Step 3: Substitute into the Determinant Formula

Substitute the values into the determinant formula: \(5(-1) - 3(6)\).

Step 4: Calculate the Determinant

Calculate the expression from Step 3: \(5(-1) - 3(6) = -5 - 18 = -23\).

Key Concepts

Matrix Operations

Matrix operations are a fundamental part of linear algebra that involve various manipulations of matrices. These operations include addition, subtraction, multiplication, and finding determinants. Each operation has specific rules:

• Addition/Subtraction: Matrices must be of the same size. Corresponding elements are added or subtracted.
• Multiplication: To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second. The resulting matrix has dimensions equal to the number of rows in the first matrix and columns in the second.
• Determinants: A number that provides important properties of a matrix, critical in solving systems of linear equations, among other applications.

Mastering these operations is key to understanding complex mathematical concepts in linear algebra.

Determinant Formula

The determinant of a matrix is a special number that can provide insights into the matrix's properties. For a simple \(2 \times 2\) matrix, the determinant is straightforward to calculate. Here's how:

• Given a matrix \(\left|\begin{array}{cc} a & b \ c & d \end{array}\right|\), the formula is \(ad - bc\).
• This formula uses the multiplying of elements on the diagonal (top left to bottom right) subtracting the product of elements on the other diagonal.

Calculating the determinant can determine if a matrix has an inverse and tell whether a system of equations has a unique solution. It's a critical tool in linear algebra.

Linear Algebra

Linear algebra is a branch of mathematics centered around vectors, vector spaces, and linear equations. Some of its core applications include:

• Solving systems of linear equations efficiently.
• Understanding geometric concepts like lines and planes in higher dimensions.
• Working with vectors to describe physical phenomena in engineering and physics.

Mastering linear algebra opens the door to advanced mathematics and numerous STEM fields. It provides the framework for more complex concepts like eigenvalues and eigenvectors.

Matrix Theory

Matrix theory examines the properties and applications of matrices in mathematics. It's an essential part of understanding more advanced mathematical and computational problems:

• Matrices are used to represent and solve linear equations.
• They are core to many algorithmic processes in computer science, from graphics to artificial intelligence.
• Matrix theory helps in exploring transformations, such as rotations and scaling in geometry.

Studying matrix theory not only helps with processing large amounts of data but also enhances problem-solving skills in numerous scientific disciplines.