Problem 16 - Advanced Engineering Mathematics

Question

In Problems 1-20, fill in the blanks or answer true/false. The augmented matrix $$ \left(\begin{array}{lll|l} 1 & 1 & 1 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right) $$ is in reduced rowechelon form._________

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understanding Reduced Row Echelon Form (RREF)

Reduced Row Echelon Form (RREF) of a matrix adheres to specific conditions: each leading entry (the first non-zero number from the left) in a row is 1, each leading 1 is the only non-zero entry in its column, and the leading 1 in each non-zero row appears to the right of the leading 1 in the row directly above. Also, any zero rows, if present, must be at the bottom.

Step 2: Analyze the Given Matrix

The given matrix is: \[\left(\begin{array}{lll|l} 1 & 1 & 1 & 2 \ 0 & 1 & 0 & 3 \ 0 & 0 & 0 & 0 \end{array}\right) \]Let's check if it meets the RREF conditions: 1. The leading entry in the first row is 1 and is the only non-zero number in its column if pivots of other rows are ignored.2. The leading entry in the second row is also 1 and it is the only non-zero number in its column.3. The third row is a zero row, appropriately placed at the bottom.

Step 3: Evaluate Columns

Column 1: The leading 1 in the first row fulfills the requirement of having all zeros below it. Column 2: The leading 1 in the second row also fulfills the requirement, as it has zeros both below and above within the reduced pivot portion. Column 3: No leading entry is not an issue as there is no requirement for every column to have one.

Step 4: Conclusion on Matrix Form

The matrix adheres to all the requirements for a reduced row echelon form. The leading entries are clean, positioned as needed, and zero rows are at the bottom. Therefore, the statement about the matrix being in RREF is true.

Key Concepts

Augmented Matrix

An augmented matrix is a way to represent a system of linear equations. It combines the coefficients of the variables and the constants from the equations into a single matrix format. This makes it easier to perform operations and solve the system using methods like Gaussian elimination or Gauss-Jordan elimination. To create an augmented matrix:

• Write down the coefficients of the variables from each equation as rows.
• Include a vertical line to separate the coefficients from the constants on the right-hand side of the equations.

In the example given, the augmented part after the vertical line represents the constants from the equations. This format is particularly useful for matrix manipulations to find solutions for the system of equations.

Leading Entry

The leading entry in a matrix is the first non-zero number in each row when moving from left to right. In reduced row echelon form (RREF), the leading entry must be 1, and each leading 1 must be the only non-zero entry in its column. Here are key things to remember about leading entries:

• Each leading 1 must appear to the right of any leading 1s in the rows above.
• Leading entries help in simplifying the matrix to solve equations by eliminating other coefficients in their columns.

In the given matrix:

• Row 1 and Row 2 both have leading entries of 1.
• These leading 1s are positioned correctly according to RREF rules.

This structure helps in systematically solving systems of equations.

Zero Row

A zero row in a matrix is any row that consists entirely of zeros. In reduced row echelon form, any zero rows must appear at the bottom of the matrix. This arrangement helps maintain the hierarchy of non-zero rows, which is important for clarity in solving systems of equations. Understanding zero rows is important because:

• They indicate that one of the equations in the system was trivial or redundant, contributing no new information.
• Ensuring zero rows are at the bottom helps in correctly applying elimination methods.

In the matrix provided, the third row is a zero row, and it is correctly placed at the bottom, meeting the requirements for reduced row echelon form.

Matrix Analysis

Matrix analysis involves examining the components and structure of a matrix to determine its properties and suitability for representing systems of equations. In terms of RREF, analysis ensures that the matrix meets all conditions such as proper placement of leading entries and zero rows. For matrix analysis, you should:

• Check that the first non-zero number in each row is 1.
• Verify leading 1s have zeros in all other entries in their columns.
• Ensure that zero rows are at the bottom of the matrix.

Effective matrix analysis can confirm the matrix's readiness to provide solutions, preventing errors in solving a system of equations. The example matrix provided meets all the RREF criteria, confirming the accuracy of the matrix analysis.