Problem 4 - Algebra for College Students

Question

Indicate whether each matrix is in reduced echelon form. \(\left[\begin{array}{lll:r}1 & 0 & 0 & 5 \\ 0 & 3 & 0 & 8 \\ 0 & 0 & 1 & -11\end{array}\right]\)

Step-by-Step Solution

Answer

The matrix is not in reduced row echelon form.

Step 1: Identify Leading Entries

In reduced row echelon form (RREF), each leading entry of a nonzero row is 1. Identify the leading entries in each row of the matrix.Given matrix:\[\begin{bmatrix}1 & 0 & 0 & 5 \0 & 3 & 0 & 8 \0 & 0 & 1 & -11\end{bmatrix}\]Leading entries are at positions (1,1), (2,2), and (3,3).

Step 2: Ensure Leading Ones

Check if all leading entries identified in Step 1 are 1. The leading entry in the second row is 3, not 1, which violates the requirement for RREF.

Step 3: Verify Column Conditions

In RREF, the column containing a leading 1 must have all other entries as zero. Check columns: Column 1 and Column 3 have leading 1s and zeros elsewhere, but Column 2 has a leading entry of 3, not 1.

Step 4: Confirm Row Order

Nonzero rows must be above any rows of zeros. This matrix has no rows of zeros, so it satisfies this condition.

Step 5: Evaluate Final Form

Given that the matrix does not satisfy the condition where each leading entry in all nonzero rows must be 1, it cannot be in reduced echelon form.

Key Concepts

Matrices

A matrix is essentially a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is widely used in different areas like physics, computer science, and mathematics. In this context, we're focusing on matrices as they relate to solving systems of linear equations and transformations.
Consider the matrix in the original problem: it has three rows and four columns, described as a "3 by 4" matrix. Each element in this matrix has a specific location described by its row and column indices. For instance, in the matrix presented in the exercise, the entry 3 is located in the second row and second column.
Matrices are a useful way to represent linear equations systematically and explore their solutions. By applying operations to matrices, we can simplify these systems or even transform them into a specific form, such as reduced echelon form, which helps in understanding the solutions better.

Row Echelon Form

The row echelon form (REF) of a matrix is one of the systematic ways to simplify matrices for solving systems of equations. To be in REF, a matrix must meet specific criteria.

• Each leading entry, or the first non-zero number from the left in each row, must be to the right of the leading entry in the row above.
• Each leading entry is a non-zero number.
• All entries in a column below a lead entry must be zero.
• Any rows that consist entirely of zeros are at the bottom of the matrix.

By using elementary row operations—such as swapping rows, multiplying or dividing rows by a constant, and adding or subtracting multiples of one row to another—we can transform any given matrix into REF. This makes it easier to understand and work with, especially when tackling linear algebra problems.

Leading Entries

Leading entries are pivotal in both row echelon forms (REF) and reduced row echelon form (RREF). They are the first non-zero numbers in each row as you proceed from left to right.
In the context of the problem, identifying the leading entries is crucial because they determine whether the matrix is in the desired form or not.
For RREF specifically:

• Leading entries must be 1. This is essential when considering the conditions for reduced forms.
• The leading 1 must be the only non-zero number in its column.
• The row containing the leading 1 should be positioned properly relative to others.

Identifying lead entries allows us to apply the right transformations to achieve the desired matrix form. In our example, though the matrix has leading entries, they don't meet all conditions required for RREF due to the non-unit value in the second row.

Matrix Conditions

To determine whether a matrix is in reduced echelon form, it needs to satisfy specific conditions:

• All leading entries must be 1.
• The leading 1 in each row must be the only non-zero entry in its column.
• If a column contains a leading 1, all other elements in that column should be zero.
• The rows are organized in order so that any row with a leading entry is above rows with no non-zero entries.

In the matrix from the exercise, the second row has a leading entry of 3, not 1, which breaks the requirement for reduced echelon form. Similarly, these row order and entry conditions help ensure matrix simplification is achieved and the system solution is transparent. Identifying the shortcomings of a matrix can guide how to correctly transform it into the reduced form.