Question

State whether or not the given matrices are in reduced row echelon form. If it is not, state why. (a) \(\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) (b) \(\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\) (c) \(\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\) (d) \(\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]\)

Short answer

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

Step-by-step solution

Understanding Reduced Row Echelon Form (RREF)

A matrix is in reduced row echelon form if it satisfies the following conditions: 1. Any rows consisting entirely of zeros (if any) are at the bottom of the matrix. 2. The first non-zero element in each non-zero row, called the leading entry, is 1. 3. Each leading 1 is the only non-zero entry in its column. 4. The leading entry of each non-zero row is to the right of the leading entry of the previous row.

Analyze Matrix (a)

Matrix (a) is \[\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]- The leading entry of the first row is 1, and it is the only non-zero entry in its column.- The leading entry of the second row is also 1, and it is the only non-zero entry in its column.- Each leading 1 is positioned to the right of the leading 1 in the previous row.- There are no zero rows, so this matrix is in reduced row echelon form.

Analyze Matrix (b)

Matrix (b) is \[\begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\]- The first row starts with 0, which prevents it from having a leading 1 as expected in RREF.- The leading 1 in the first row should be further left than the leading 1 in the second row, which is not the case here. - Therefore, this matrix is not in reduced row echelon form.

Analyze Matrix (c)

Matrix (c) is \[\begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}\]- The first row has a leading 1, but the second row has a leading 1 directly under an existing leading 1 from the first row.- To be in RREF, the leading entries should not align vertically in the same column.- This matrix is not in reduced row echelon form.

Analyze Matrix (d)

Matrix (d) is \[\begin{bmatrix} 1 & 0 & 1 \ 0 & 1 & 2 \end{bmatrix}\]- The first row has a leading 1, and it satisfies the requirement of being the only non-zero entry in its column.- The second row also has a leading 1, which is the only non-zero entry in its column.- The leading entry of the second row is reduced correctly with respect to the leading entry of the previous rows.- This matrix satisfies all the conditions for reduced row echelon form.

Key concepts

Matrix Conditions

To determine if a matrix is in Reduced Row Echelon Form (RREF), it must meet several specific conditions. These conditions serve as guidelines that shape the structure and interpretation of the matrix in linear algebra.

• Though RREF is a concept in matrices that may seem complex initially, having a checklist of conditions helps immensely. First, any row in the matrix that is entirely zeros must appear at the bottom. This ensures that all non-zero rows are prioritized for analysis.
• A leading entry 1 in a non-zero row should be the leading entry of that row, but it must also be shifted further to the right than any leading 1 in the rows above it.
• Additionally, each leading 1 must be the only non-zero type in its column. Hence, no other non-zero elements may exist in the column, ensuring a clean linear hierarchy of rows.

These conditions make navigating and using matrices more structured, allowing clearer insights into solutions and outcomes.

Leading Entry

The leading entry in a matrix is the first non-zero number encountered as you move from left to right across a row. In Reduced Row Echelon Form, having a leading entry of 1 is crucial.

• This leading 1 sets the stage for determining pivotal positions in row operations, distinguishing critical points within the solution space of a matrix equation.
• For effective matrix computations, this leading entry of 1 must be carefully managed to be the sole non-zero entry in its entire column to maintain the clarity of the matrix's reduced form.

Understanding the role of leading entries simplifies complex matrix manipulations and fosters a streamlined approach to linear algebra operations

Zero Rows

Zero rows within a matrix contain only zeros. These rows signify no contribution to the solutions of matrix equations. Therefore, in the context of Reduced Row Echelon Form, zero rows are always a feature to locate at the bottom of the matrix.

• Positioning zero rows at the bottom aligns with the convention of RREF, as it provides a systematic way to reveal meaningful data in the matrix. This positioning aids in clarity and reduces visual confusion when interpreting a matrix structurally.
• Such rows do not affect the linear independence of the set and often represent redundant information or constraints within constraints, depending on the context of the system being analyzed.

This understanding is pivotal for both manual manipulations and computational algorithms designed to simplify matrix solutions.

Matrix Analysis

When analyzing a matrix to determine if it is in Reduced Row Echelon Form, careful attention is necessary to avoid common pitfalls. Here’s a step-by-step method:

• First, scan through each row to identify leading 1 entries, verifying that they exist and are isolated in their respective columns.
• Check the order of these leading 1s; they must ascend left to right and from top to bottom. If any 1 appears in the same position or below another in a previous row, the matrix doesn't qualify.
• Inspect zero rows last, ensuring they are positioned at the bottom—this particular hierarchy makes it easier to assess column and row priorities effectively.

By following this structured analysis, you can quickly ascertain the compliance of a matrix with the reduced form requirements, facilitating faster and more reliable problem-solving in linear algebra.