Problem 8 - Differential Equations and Linear Algebra

Question

determine whether the given matrices are in reduced row-echelon form, row- echelon form but not reduced row-echelon form, or neither. $$\left[\begin{array}{llll} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$$.

Step-by-Step Solution

Answer

The given matrix \(\left[\begin{array}{llll} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]\) satisfies all conditions for both row-echelon form and reduced row-echelon form, as all the rows are zero rows. Therefore, the matrix is in both row-echelon form and reduced row-echelon form.

Step 1: Identify all zero rows

In this matrix, all the rows are zero rows, which means that condition 1 for both row-echelon form and reduced row-echelon form is satisfied.

Step 2: Check for the leading entry of each non-zero row

Since all rows in the matrix are zero rows, there are no non-zero rows. Therefore, conditions 2 and 3 don't apply for this matrix.

Step 3: Check for pivot columns in the reduced row-echelon form

As there are no non-zero rows, there are no pivot columns, so condition 4 is trivially satisfied. Based on the properties of row-echelon form and reduced row-echelon form discussed above, we can conclude that this matrix is in both row-echelon form and reduced row-echelon form.

Key Concepts

Zero Rows

Zero rows in a matrix are simply rows where all the elements are 0. These rows do not carry any additional information and are quite special when considering matrix forms like echelon forms. In any form, be it row-echelon or reduced row-echelon, zero rows are usually pushed to the bottom. This is because they do not affect calculations such as the determinant or solutions to linear equations.
In matrices:

• Zero rows indicate a kind of neutrality. They don't bring new information to the set of equations represented by the matrix.
• They can make it easier to identify essential features in the non-zero parts of the matrix.
• When converting a matrix to its row-echelon form, it is important to push zero rows down to maintain consistency of form.

In many cases, like the given matrix in this exercise, all rows might be zero. In such cases, determining the form of the matrix could be relatively straightforward because neither non-zero rows nor pivot columns are present.

Row-Echelon Form

The row-echelon form of a matrix is considered a crucial step when solving linear equations or analyzing matrix properties. It has specific characteristics that set it apart:

• All zero rows, if present, must be at the bottom of the matrix.
• Each leading non-zero element, also known as a leading entry, must be to the right of the leading entry in the row above.
• Each column that contains a leading entry has only zeros below it.

In matrices comprised entirely of zero rows, as in the exercise, these specific criteria might seem irrelevant because no nonzero leading entries exist to dictate form. However, these matrices still qualify as being in row-echelon form, as they trivially meet the conditions where zero rows lie at the bottom and no non-zero rows exist to compare leading entries.

Pivot Columns

Pivot columns are crucial in both row-echelon and reduced row-echelon forms for describing the makeup and solution set of a matrix of equations. A pivot column contains at least one leading entry, which is the first non-zero number from left to right in its row, serving as a beacon to indicate progress through that matrix.
To identify pivot columns:

• Look for the leading entry or pivot in each row, this entry determines the column as a pivot column.
• Remember, in reduced row-echelon form, each pivot is the only non-zero entry in its column, emphasizing simplicity and clarity.

In the exercise matrix, which consists solely of zero rows, there are no pivots, and thus no pivot columns. The absence of pivot columns may suggest an overdetermined system with dependent equations, yet in an entirely zero-matrix, these conclusions are more moot than revealing. This characteristic, however, ensures the matrix satisfies conditions for both row-echelon form and reduced row-echelon form.