Problem 20 - Precalculus with Limits

Question

In Exercises 5-20, find the determinant of the matrix. \(\left[ \begin{array}{r} \frac{2}{3} & \frac{4}{3} \\ -1 & -\frac{1}{3} \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the given matrix is \( \frac{10}{9} \).

Step 1: Identify the elements of the matrix

First, identify the elements of the 2x2 matrix. They are: \( a = \frac{2}{3} \), \( b = \frac{4}{3} \), \( c = -1 \), and \( d = -\frac{1}{3} \).

Step 2: Apply the formula for the determinant of a 2x2 matrix

The formula for the determinant of a 2x2 matrix is \( ad - bc \). Substitute the identified matrix elements into the formula: \( (\frac{2}{3})(-\frac{1}{3}) - (-1)(\frac{4}{3}) \).

Step 3: Simplify the expression

Simplify the expression to find the determinant. It simplifies to: \( -\frac{2}{9} - -\frac{4}{3} = -\frac{2}{9} + \frac{4}{3} = \frac{10}{9} \).

Key Concepts

2x2 matrix

A 2x2 matrix is a simple and fundamental concept in linear algebra. It consists of four elements arranged in two rows and two columns. In the given exercise, the matrix is represented as:\[\begin{bmatrix}\frac{2}{3} & \frac{4}{3} \-1 & -\frac{1}{3}\end{bmatrix}\]Each of these elements has a specific role in various matrix operations, including the calculation of the determinant. The rows and columns facilitate handling multiple linear equations using compact notation. This form also makes it easy to apply standard mathematical operations, helping in solving diverse algebraic problems.

linear algebra

Linear algebra is a branch of mathematics focusing on vectors, vector spaces, and linear transformations. It provides the framework for analyzing linear systems through matrices, which are arrays of numbers arranged in rows and columns. In linear algebra, matrices are used to represent linear equations and transformations efficiently. By using matrix operations, one can find solutions to systems of linear equations, describe geometric transformations, and perform operations on vectors with ease. One crucial aspect of linear algebra is the determinant, which provides essential information about a matrix. Determinants help in assessing the invertibility of matrices and are vital in solving equations where matrices transform vector spaces.

matrix operations

Matrix operations are techniques used to manipulate matrices and perform calculations that solve complex mathematical problems. Some common operations include addition, subtraction, multiplication, and finding the determinant.

• Addition and Subtraction: Matrices of the same size can be added or subtracted by adding or subtracting their corresponding elements.
• Multiplication: There are different types of multiplication, including multiplying a matrix by a scalar or another matrix, following specific rules for element calculation.
• Determinant: The determinant is a scalar value derived from a square matrix. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), its determinant is computed as \( ad - bc \).

Through matrix operations, a multitude of mathematical and real-world problems can be solved more systematically. The determinant, specifically, provides insights into the properties of matrices, such as invertibility and the ability to describe unique solutions for a system of equations.