Problem 11 - College Algebra with Modeling and Visualization

Question

Is the matrix in row-echelon form? (a) \(\left[\begin{array}{ll|r}1 & 3 & 2 \\ 0 & 1 & -1\end{array}\right]\) (b) \(\left[\begin{array}{rrr|r}1 & 4 & -1 & 0 \\ 0 & -1 & 1 & 3 \\ 0 & 2 & 1 & 7\end{array}\right]\) (c) \(\left[\begin{array}{rrr|r}1 & 6 & -8 & 5 \\ 0 & 1 & 7 & 9 \\ 0 & 0 & 1 & 11\end{array}\right]\)

Step-by-Step Solution

Answer

(a) Yes, (b) No, (c) Yes

Step 1: Identify row-echelon form criteria

For a matrix to be in row-echelon form, it must satisfy the following criteria: 1. All nonzero rows are above any rows of all zeros. 2. The leading coefficient (the first nonzero number from the left, also called a pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it. 3. The leading coefficient in any nonzero row is 1 and the columns containing a leading 1 have zeros below them.

Step 2: Analyze matrix (a)

Matrix (a) is \[\begin{bmatrix} 1 & 3 & 2 \ 0 & 1 & -1 \end{bmatrix}\]- Row 1 has a leading 1 at column 1. - Row 2 has a leading 1 at column 2, which is to the right of Row 1's leading 1. - There are no rows of all zeros. These satisfy all the criteria for row-echelon form.

Step 3: Analyze matrix (b)

Matrix (b) is \[\begin{bmatrix} 1 & 4 & -1 & 0 \ 0 & -1 & 1 & 3 \ 0 & 2 & 1 & 7 \end{bmatrix}\]- Row 1 has a leading 1 at column 1.- Row 2's leading coefficient (nonzero) is -1 at column 2. This is in the correct position if it was positive 1 but does not satisfy the condition of the leading coefficient being 1. - Additionally, below this pivot, Row 3 has a nonzero entry in column 2 (which should be zero below the pivot in a fully reduced echelon form).Matrix (b) does not satisfy row-echelon form conditions.

Step 4: Analyze matrix (c)

Matrix (c) is \[\begin{bmatrix} 1 & 6 & -8 & 5 \ 0 & 1 & 7 & 9 \ 0 & 0 & 1 & 11 \end{bmatrix}\]- Row 1 has a leading 1 at column 1.- Row 2 has a leading 1 at column 2, which is to the right of Row 1's leading 1. - Row 3 has a leading 1 at column 3, which is to the right of Row 2's leading 1. - The leading 1s in columns 1, 2, and 3 have zeros below them.- There are no rows of all zeros. All conditions are satisfied, so matrix (c) is in row-echelon form.

Key Concepts

Matrices

Matrices are an indispensable tool in mathematics, particularly in linear algebra. They are essentially rectangular arrays of numbers arranged in rows and columns. Each position in a matrix is composed of entries that represent specific values. Matrices can vary in size, denoted as "m x n," where "m" represents the number of rows and "n" denotes the number of columns.

Matrices serve as a compact way to represent linear equations and transformations. They can represent everything from systems of equations to transformations in geometry. When you line up several linear equations, the coefficients form a matrix. For example, a system of equations like:
- 3x + 2y = 7
- x - 4y = -2
can be succinctly represented by the matrix \(\begin{bmatrix} 3 & 2 \ 1 & -4\end{bmatrix}\).

The versatility of matrices extends to operations like addition, subtraction, and most importantly, multiplication, which aids in solving systems of equations or transforming geometric objects. One key aspect of matrix multiplication is that it is not commutative, meaning that the order matters: \(AB eq BA\) in most cases.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on the study of vectors, vector spaces (also known as linear spaces), and linear transformations. It's all about understanding the properties and applications of lines and planes, and by extension, mathematical structures like matrices.

In linear algebra, matrices play a crucial role in solving linear systems, understanding transformations, and representing data. One important concept in linear algebra is the row-echelon form of a matrix. This specific arrangement makes it easier to solve systems of linear equations or even determine the matrix's rank, which tells you about the solutions' uniqueness or existence.

Linear algebra is considered the backbone of many mathematical applications in different fields such as computer science, engineering, physics, economics, and more. Understanding how to manipulate matrices and use them to perform various operations opens up a wide range of possibilities in abstract and practical problem-solving.

Matrix Operations

Matrix operations are fundamental procedures that allow us to manipulate and solve problems involving matrices. The most common operations include addition, subtraction, multiplication, and finding inverses. Each of these operations has specific rules and requirements that need to be adhered to for successful computation.

- **Addition/Subtraction**: Two matrices can be added or subtracted only if they are of the same size. You perform the operation element-wise, adding or subtracting corresponding elements.

- **Matrix Multiplication**: Unlike addition, multiplication is not done element-wise. Instead, the rows of the first matrix are multiplied with the columns of the second matrix. This operation requires that the number of columns in the first matrix is equal to the number of rows in the second one.

- **Inversion**: A matrix inverse can only be found for a square matrix that is non-singular (i.e., has a non-zero determinant). The inverse of a matrix, when multiplied with the original matrix, results in the identity matrix.

One specific operation related to row-echelon form is the process of row reduction, which systematically changes the matrix into a form that is easier to work with, often used in the solution of linear systems. Mastering matrix operations is indispensable for tackling problems not only in mathematics but also in various fields where data and relationships are modelled and analyzed using matrices.