Problem 24 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{cc} -7 & -7 \\ -6 & -4 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is -14.

Step 1: Determine the Formula for a 2x2 Determinant

To evaluate the determinant of a 2x2 matrix \[ \left|\begin{array}{cc} a & b \ c & d \end{array}\right| \]use the formula: \[ \text{Determinant} = ad - bc \]where the elements of the matrix are:\[ \left|\begin{array}{cc} a & b \ c & d \end{array}\right| \].

Step 2: Identify the Elements of the Matrix

From the given matrix:\[ \left|\begin{array}{cc} -7 & -7 \ -6 & -4 \end{array}\right| \]we identify the elements:\( a = -7 \),\( b = -7 \), \( c = -6 \), and \( d = -4 \).

Step 3: Substitute the Elements into the Formula

Using the formula \( ad - bc \), substitute the values:\[ \text{Determinant} = (-7)(-4) - (-7)(-6) \]

Step 4: Calculate Each Product

First calculate \((-7)(-4)\): \[ (-7) \times (-4) = 28 \]Next, calculate \((-7)(-6)\): \[ (-7) \times (-6) = 42 \].

Step 5: Subtract the Products

Subtract the second product from the first product:\[ 28 - 42 = -14 \]This gives the value of the determinant.

Key Concepts

Matrix Algebra

Matrix algebra is an essential part of linear algebra that deals with the study of matrices and their operations. It provides a systematic method for solving systems of linear equations and is widely used in various fields, such as physics, engineering, and computer science. Matrices are rectangular arrays of numbers, which can represent data and linear transformations. In matrix algebra, you can perform operations like addition, subtraction, multiplication, and finding the determinant, which is the focus of this exercise.

When working with matrix algebra, it is crucial to understand the properties of different types of matrices and their respective operations. For instance:

• Diagonal Matrices: Matrices that have all off-diagonal elements as zero.
• Identity Matrices: Square matrices with ones on the diagonal and zeros elsewhere.
• Square Matrices: Matrices with the same number of rows and columns.

These matrices have special properties that make calculations more straightforward, particularly when multiplying matrices or finding determinants.
To effectively use matrix algebra, it's important to master the basic operations, understand the geometric interpretations of matrices, and develop a problem-solving mindset to apply these concepts in real-world situations.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix in linear algebra, consisting of two rows and two columns. Its general representation looks like the following:\[\begin{pmatrix} a & b \ c & d \end{pmatrix}\]
In this matrix, \(a\), \(b\), \(c\), and \(d\) represent the elements of the matrix, which can be any real numbers. A major reason why 2x2 matrices are popular in teaching is that they lay the groundwork for understanding more complex matrices. Moreover, they provide a simple yet powerful way to visualize linear transformations in a 2-dimensional space.

When dealing with determinants, a specific formula allows you to compute the determinant of a 2x2 matrix effortlessly. The formula is:\[\text{Determinant} = ad - bc\]
This formula arises from the concept of linear transformations represented by the matrix and can be used to determine properties such as whether the matrix is invertible. For example, if the determinant of a matrix is zero, the matrix does not have an inverse and corresponds to a case where the transformation it represents leads to a collapse in space (like mapping a plane to a line). This fundamental property of the determinant in 2x2 matrices makes it a key focus in both theoretical exploration and practical application.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on vector spaces, linear transformations, and matrices. It is a foundational subject in mathematics, providing the tools necessary to manipulate and solve linear equations. The study of linear algebra involves understanding concepts such as vector spaces, basis, dimension, and linear independence.

• Vector Spaces: These are collections of vectors that can be added together and multiplied by scalars while satisfying specific properties.
• Linear Transformations: Functions that map from one vector space to another while preserving vector addition and scalar multiplication. These can be represented by matrices.
• Determinants: A scalar value associated with a square matrix, providing important information about the matrix, such as whether it is invertible.

Linear algebra is vital in various applications, including computer graphics where transformations like rotation and scaling are expressed using matrices, data analysis where dimensionality reduction techniques use eigenvalues and eigenvectors, and in solving large systems of linear equations in engineering and physics.
Understanding the core aspects of linear algebra opens up a world of mathematical modeling possibilities, serving as a bridge to more advanced topics and a plethora of real-world applications where linear models are vital.