Question

State whether or not the given matrices are in reduced row echelon form. If it is not, state why. (a) \(\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]\) (b) \(\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 1\end{array}\right]\) (c) \(\left[\begin{array}{lll}0 & 0 & 0 \\ 1 & 0 & 0\end{array}\right]\) (d) \(\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]\)

Short answer

(a) Yes, in RREF. (b) No, incorrect leading entries rules. (c) No, zero row not at bottom. (d) Yes, in RREF.

Step-by-step solution

Definition Review

Reduced Row Echelon Form (RREF) is characterized by the following criteria: 1. Any row consisting entirely of zeros is at the bottom of the matrix. 2. The leading entry (first non-zero number from the left) of each non-zero row is 1. 3. The leading entry of a row is in a column to the right of the leading entry of the above row. 4. Each leading 1 is the only non-zero entry in its column.

Analyze Matrix (a)

Matrix (a) is given by \(\left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 0 & 1\end{array}\right]\).- The leading entry of the first row is 1.- The leading entry of the second row is also 1, located in a column to the right of the leading entry in the first row.- Each leading 1 is the only non-zero entry in its column.- There are no rows of all zeros.Therefore, matrix (a) is in Reduced Row Echelon Form.

Analyze Matrix (b)

Matrix (b) is given by \(\left[\begin{array}{ccc}1 & 0 & 1 \ 0 & 1 & 1\end{array}\right]\).- The leading entry of the first row is 1, and the leading entry of the second row is also 1, appearing in a farther column to the right.- Each leading 1 is the only non-zero entry in its column; however, in the first column, the non-leading entry should be 0.Therefore, matrix (b) is not in Reduced Row Echelon Form because the first column has a non-zero entry above the leading 1 of column 3.

Analyze Matrix (c)

Matrix (c) is given by \(\left[\begin{array}{ccc}0 & 0 & 0 \ 1 & 0 & 0\end{array}\right]\).- The first row is all zeros but appears above a non-zero row.This matrix is not in Reduced Row Echelon Form because the zero row is not at the bottom.

Analyze Matrix (d)

Matrix (d) is given by \(\left[\begin{array}{ccc}0 & 0 & 0 \ 0 & 0 & 0\end{array}\right]\).- All rows are zero.Therefore, matrix (d) is in Reduced Row Echelon Form because all-zero rows are allowed at the bottom of the matrix.

Key concepts

Matrix Analysis

Matrix Analysis is the examination of matrices to understand their properties and transformations. A matrix is a rectangular array of numbers. It's essential in various fields like physics, computer science, and geometry.
The process of analyzing matrices in this context involves checking if a matrix complies with specific criteria like the Reduced Row Echelon Form (RREF). This form ensures matrices are simplified for solving linear equations.

• In RREF, each leading entry of a row is a 1 and is the only non-zero entry in its column.
• Zero rows must be below any non-zero rows.
• Leading 1s in subsequent rows appear further to the right than those in previous rows.

This form is unique for every matrix and is invaluable for understanding system solutions. Analyzing matrices allows us to simplify matrix equations and uncover important characteristics.

Linear Algebra

Linear Algebra is a branch of mathematics that focuses on vector spaces and linear mappings between these spaces. It's fundamentally about solving systems of linear equations, which are frequently represented with matrices.
Matrices, in linear algebra, represent linear transformations. Analyzing their form—such as whether they're in Reduced Row Echelon Form—can simplify finding solutions and understanding the behavior of mathematical models.
Key aspects include:

• Understanding matrices and determinants.
• Vector operations and linear transformations.
• Using matrices to solve linear systems efficiently.

The simplification of matrices through techniques like putting them in RREF helps solve linear equations easily. Linear algebra provides the tools necessary to manipulate these matrices, making it a foundational element for advanced mathematics and applications.

Row Operations

Row Operations are methods to manipulate the rows of a matrix to achieve a desired form like the Reduced Row Echelon Form. These operations include row swapping, scaling rows, and adding or subtracting rows. These are crucial for reaching a form that reveals solutions to linear equations.
Some basic row operations are:

• Row swapping: Exchanging two rows within the matrix.
• Row scaling: Multiplying all elements in a row by a non-zero scalar.
• Row addition: Adding or subtracting a multiple of one row to another row.

These operations are fundamental for matrix manipulation because they don't change the solutions of the linear equations represented by the matrix. By performing these operations, we can transform any matrix into its simplest form making it easier to identify potential solutions.