Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, find the determinant. $$ \left|\begin{array}{rrr} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -3 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 3.

Step 1: Understanding the Determinant of a Diagonal Matrix

When dealing with a diagonal matrix, where all the non-diagonal elements are zero, the determinant is the product of the diagonal elements. This makes calculating determinants for diagonal matrices straightforward.

Step 2: Identifying the Diagonal Elements

Look at the matrix and identify the elements on the main diagonal, which are the ones at positions (1,1), (2,2), and (3,3). In this matrix, the diagonal elements are -1, 1, and -3.

Step 3: Multiplying the Diagonal Elements

Calculate the determinant by multiplying the diagonal elements together: \[-1 imes 1 imes (-3)\] This simplifies to -1 times 1 times -3 which equals 3.

Step 4: Concluding the Problem

Since the determinant is the product of the diagonal elements, for this diagonal matrix, the determinant is 3.

Key Concepts

Diagonal Matrix

A diagonal matrix simplifies many mathematical operations in linear algebra. This special type of square matrix has all its elements outside the main diagonal equal to zero. Having this property brings about several conveniences, especially when calculating the determinant.
An example of a diagonal matrix is:
\[ \begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix} \]
Here, \( a \), \( b \), and \( c \) are the diagonal elements, and they can be any real numbers.

• If a matrix is diagonal, the determinant can be found simply by multiplying these diagonal entries together.
• This property makes diagonal matrices particularly useful in simplifying more complex matrix operations, allowing quick calculations without resorting to more extensive algebraic processes.

Diagonal matrices are not only present in pure mathematics but also have applications in computer algorithms and data analysis, where data simplification is often needed.

Matrix Operations

Matrix operations are the foundation of many computations in linear algebra. There are several basic operations associated with matrices, such as addition, subtraction, scalar multiplication, matrix multiplication, and others. Each operation follows specific rules and is essential in a variety of mathematical fields.

• Addition and Subtraction: Two matrices of the same dimension can be added or subtracted by performing the operation element-wise. That means you simply add or subtract corresponding elements.
• Scalar Multiplication: This involves multiplying every cell in the matrix by the same number (the scalar), effectively scaling the matrix.
• Matrix Multiplication: This more complex operation requires the number of columns in the first matrix to match the number of rows in the second. It involves multiplying elements and adding the resulting products, which can produce a new matrix with different dimensions.

Each of these operations is crucial for solving systems of equations, transforming geometric figures, and more. Understanding these basic operations is essential to advancing in linear algebra.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. It is fundamental to both pure and applied mathematics. Linear algebra allows for the understanding and manipulation of data structures, providing a framework for modeling many real-world systems.
Key elements of linear algebra include:

• Vectors: These are objects that represent both magnitude and direction, often used to signify a point in space.
• Vector Spaces: Collections where vectors can be added together and multiplied by scalars while still remaining within the space.
• Linear Transformations: These are mappings between vector spaces that preserve the operations of addition and scalar multiplication.

Beyond the theoretical aspects, linear algebra is incredibly important in fields like computer science, engineering, physics, statistics, and data science. Its concepts form the basis for algorithms in machine learning, optimizations, and simulations. Thus, a strong grasp of linear algebra is essential for anyone looking to understand and develop modern scientific models.