Problem 8 - Algebra for College Students

Question

Evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{ll}-4 & -3 \\ -5 & -7\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant of the matrix is 13.

Step 1: Understanding the Determinant Formula

For a given \(2 \times 2\) matrix, the determinant can be calculated using the formula \(\text{det}(A) = ad - bc\). Here, \(a, b, c,\) and \(d\) are elements of the matrix arranged as follows: \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\).

Step 2: Identify Elements of the Matrix

Given the matrix \(\begin{pmatrix} -4 & -3 \ -5 & -7 \end{pmatrix}\), identify the elements as follows: \(a = -4\), \(b = -3\), \(c = -5\), and \(d = -7\).

Step 3: Apply the Determinant Definition

Using the formula \(ad - bc\), substitute the identified values: \((-4)(-7) - (-3)(-5)\).

Step 4: Perform the Calculations

Calculate each part of the expression separately: 1. \((-4)(-7) = 28\) 2. \((-3)(-5) = 15\)

Step 5: Compute the Result

Subtract the second product from the first: \(28 - 15 = 13\). Therefore, the determinant of the matrix is 13.

Key Concepts

Matrix Algebra

Matrix algebra is a specialized branch of mathematics that focuses on arrays of numbers - these arrays are called matrices. It plays a crucial role in various areas such as engineering, physics, computer science, and more.
Matices are used to solve systems of equations, transform mathematical objects, and describe linear transformations. Here are some concepts and operations in matrix algebra:

• Addition and Subtraction: Matrices can be added or subtracted when they have the same dimensions.
• Multiplication: There are two types of multiplication: scalar multiplication and matrix multiplication.
• Transpose: Flipping a matrix over its diagonal.
• Inverse: The matrix that, when multiplied with the original matrix, results in the identity matrix.

Determinants, which will be further discussed, are a fundamental attribute of square matrices and are pivotal in solving linear equations and other matrix operations.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices, consisting of two rows and two columns. It looks like this:
\(\begin{pmatrix} a & b \ c & d \end{pmatrix}\).
Each position in the matrix holds a specific function or purpose depending on the operation being performed.
For instance, elements "a" and "d" form the main diagonal of the matrix, whereas "b" and "c" sit across the other slots. These positions are important when calculating the determinant of a 2x2 matrix. Understanding the structure of a matrix is essential because it determines how calculations are carried out. The 2x2 matrix serves as a stepping stone to understanding more complex matrices. It is easy to visualize and forms the building block for larger matrices.

Determinant Calculation

The determinant is a special number that is calculated from a square matrix. For a 2x2 matrix given by \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant can be calculated using the formula:

• \(det(A) = ad - bc\)

This calculation is straightforward and involves multiplying the elements of the main diagonal ("a" and "d") and then subtracting the product of the off-diagonal elements ("b" and "c").
For example, to determine the determinant of the matrix \(\begin{pmatrix} -4 & -3 \ -5 & -7 \end{pmatrix}\):
1. Multiply the elements of the main diagonal:

• \((-4)(-7) = 28\)

2. Next, multiply the elements of the other diagonal:

• \((-3)(-5) = 15\)

3. Finally, subtract the second product from the first:

• \(28 - 15 = 13\)

Therefore, the determinant of the matrix is 13.The determinant provides important information about a matrix, such as whether it is invertible or not.

Linear Algebra

Linear algebra is a branch of mathematics that studies vectors, matrices, and linear transformations. It is widely applicable in various fields such as computer graphics, economics, and natural sciences. Within linear algebra, matrices and their determinants are foundational concepts.
Determinants of matrices are useful in several ways:

• They help in understanding the properties of a matrix, such as invertibility.
• Determinants are used to find solutions to systems of linear equations.
• They can determine whether a set of vectors is linearly independent or dependent.

Matrix algebra expands on these concepts, providing powerful techniques for solving complex mathematical problems involving linear equations and transformations. Linear algebra's primary focus is on systems of equations and transformations that can be described with linear equations, making it a cornerstone of mathematical applications in the real world.