Question

State whether or not the given matrices are in reduced row echelon form. If it is not, state why. (a) \(\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right]\) (b) \(\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{array}\right]\) (c) \(\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right]\) (d) \(\left[\begin{array}{cccc}1 & 0 & 0 & -5 \\ 0 & 1 & 0 & 7 \\ 0 & 0 & 1 & 3\end{array}\right]\)

Short answer

Matrices (a), (b), and (d) are in RREF. Matrix (c) is not.

Step-by-step solution

Define Reduced Row Echelon Form (RREF) Criteria

A matrix is in reduced row echelon form if it satisfies the following conditions: 1. Any row containing non-zero elements has 1 as its first non-zero entry (leading 1). 2. Each leading 1 is the only non-zero entry in its column. 3. The leading 1 in any row is to the right of the leading 1 in the row above it. 4. Rows with zero entries are at the bottom of the matrix.

Analyze Matrix (a)

Matrix (a) is given by:\[\begin{bmatrix}1 & 1 & 1 \0 & 1 & 1 \0 & 0 & 1\end{bmatrix}\]- Each row has a leading 1 and all elements below these leading ones are 0.- All leading 1s are to the right of the leading 1s above them.- There are no zero rows at the top.- Each leading 1 is the only non-zero entry in its column.Thus, matrix (a) is in RREF.

Analyze Matrix (b)

Matrix (b) is:\[\begin{bmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 0\end{bmatrix}\]- Leading 1 in each row is the only non-zero in its column.- Leading 1 in row 2 is to the right of the leading 1 in row 1.- Zero row is at the bottom.Therefore, matrix (b) is in RREF.

Analyze Matrix (c)

Matrix (c) is:\[\begin{bmatrix}1 & 0 & 0 \0 & 0 & 1 \0 & 0 & 0\end{bmatrix}\]- The leading 1 in row 2 is properly placed to the right compared to row 1.- However, the leading 1 in row 1 does not precede it because it is in the same column as row 3, which makes row 2 invalid.Thus, matrix (c) is not in RREF because the order of leading 1s is incorrect or does not satisfy the row condition.

Analyze Matrix (d)

Matrix (d) is:\[\begin{bmatrix}1 & 0 & 0 & -5 \0 & 1 & 0 & 7 \0 & 0 & 1 & 3\end{bmatrix}\]- Each leading 1 is the only non-zero entry in its column and each leading 1 is to the right of the leading 1 in the row above.- The leading 1s have non-zero values in other columns, which does not affect RREF statusHence, matrix (d) is in RREF.

Key concepts

Matrix Analysis

When analyzing matrices to determine if they are in reduced row echelon form (RREF), we're checking specific criteria. The structure and properties of a matrix tell us a lot about its form and the ease with which we can solve systems of linear equations or perform other operations. In essence, matrix analysis is about breaking down the matrix into understandable parts and confirming that they meet the necessary mathematical properties. By looking at the position of leading entries, examining zero rows, and verifying matrix conditions, we can effectively analyze their form. Matrix analysis forms the backbone for understanding how matrices can represent systems of equations and be manipulated to find solutions efficiently.

Leading Entries

Leading entries in a matrix are crucial when determining the form and properties of a matrix. A leading entry is the first non-zero element in a row that comes from left to right. This element is significant because:

• It signifies where a new pivot starts.
• It directs the hierarchy of rows in a matrix, showing which equations are independent in a system.
• Each leading 1 (pivot) helps in simplifying the matrix and reducing other elements to zero, crucial for achieving the simplest form of a matrix.

In order for a matrix to be in RREF, every leading 1 not only must be the first non-zero element in its row but also should be to the right of any leading entries in rows above it. This helps ensure that each subsequent row introduces a new independent variable or equation. Leading entries thus maintain the structure crucial for solving systems of equations by providing a clear pathway for simplification and solution finding.

Matrix Conditions

Matrix conditions that must be satisfied for a matrix to be in reduced row echelon form are quite specific:

• The first non-zero number in each row must be 1, referred to as a leading 1.
• Leading 1s must be the only non-zero entries in their respective columns, ensuring the simplicity of transformation.
• The leading 1 of a given row should be positioned to the right of the leading 1 in the previous row. This arrangement creates a staircase-like pattern that is pivotal to the RREF structure.
• All rows consisting entirely of zeros should appear at the bottom of the matrix, allowing for an easier reach to solutions starting from non-zero equations.

Satisfying these conditions simplifies the process of solving linear equations and ensures a unique solution, if it exists. If any one of these conditions is not met, the matrix is not in RREF and may require additional row operations to reach the desired form.

Zero Rows

In matrix analysis, zero rows represent rows where every element is zero. These rows are important because:

• They do not contribute any new information to the system of equations represented by the matrix.
• In reduced row echelon form, zero rows are placed at the bottom of the matrix. This arrangement maintains clarity and order as it shows that the non-zero rows contain all the essential information.
• Zero rows signify dependent equations in a system, meaning one or more equations may be a linear combination of others.

When examining matrices, always check that zero rows are at the bottom. This helps ensure that the matrix is ready for analysis and that solution steps, like back substitution in solving systems, can proceed without unnecessary complications. Proper placement and understanding of zero rows are essential parts of mastering matrix manipulation and simplification.