Problem 14 - Algebra and Trigonometry

Question

A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$ \left[\begin{array}{llllll} 1 & 3 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{array}\right] $$

Step-by-Step Solution

Answer

(a) Yes, in row-echelon form. (b) No, not in reduced row-echelon form. (c) Equations: \(x_1 + 3x_2 = 0\), \(x_2 = 0\), \(x_5 = 2\).

Step 1: Check Leading Entry

Examine each row from top to bottom to ensure each leading entry (first non-zero entry) is 1 and is further right than the leading entry in the row above. In the given matrix, the leading entries are 1 (1st row), 1 (2nd row), and 1 (4th row) which adhere to this criteria.

Step 2: Verify Zero Rows

Ensure that any rows that consist entirely of zeros are at the bottom of the matrix. In this matrix, there are no rows consisting entirely of zeros, so this condition is naturally met.

Step 3: Determine if Row-Echelon Form

Since each leading entry is 1 and there are no zero rows above rows with non-zero entries, the matrix satisfies the conditions for row-echelon form.

Step 4: Check Reduced Row-Echelon Criteria

In reduced row-echelon form, each leading 1 must be the only non-zero entry in its column. Check columns 1, 2, and 4. The 3rd column has no leading one, thus no check is required. Columns 1 and 2 meet this condition, but column 4 does not because of the third and fourth rows both having non-zero entries.

Step 5: Identify Matrix as Not Fully Reduced

Because the leading 1 in column 4 appears in both the 3rd and 4th row, the matrix is not in reduced row-echelon form. Further row operations are required to meet this condition.

Step 6: Translate Matrix to System of Equations

Use the augmented matrix to write the equivalent system of equations. The columns correspond to the coefficients of variables. Translate each row of the matrix into an equation:- Row 1: \( x_1 + 3x_2 + x_4 = 0 \)- Row 2: \( x_2 + 4x_4 = 0 \)- Row 3: \( x_4 + x_5 = 2 \)- Row 4: \( x_4 = 0 \)

Step 7: Simplify the System Using Information

From Row 4's equation \( x_4 = 0 \), substitute into the other equations:- Row 1 becomes \( x_1 + 3x_2 = 0 \)- Row 2 becomes \( x_2 = 0 \)- Row 3 becomes \( x_5 = 2 \)The consistent solution corresponds to the simplicity gathered from the system after substituting known values.

Key Concepts

Reduced Row-Echelon Form

In linear algebra, matrices are often transformed to the reduced row-echelon form (RREF) to simplify solving systems of equations. There’s a specific set of conditions a matrix must meet to be in RREF:

1. **Leading Entries**:
- Each leading entry (first non-zero entry) in a row is 1.
- These ones must appear in columns to the right of the leading ones in rows above.

2. **Column Conditions**:
- In each column containing a leading 1, all other entries must be zero.

3. **Zero Rows**:
- Any row with all zeros must be at the bottom of the matrix.

The given matrix does not meet all criteria for RREF since the column containing the leading 1 in the third and fourth rows also contains other non-zero entries. Therefore, it is only in row-echelon form. Performing more row operations, like subtracting rows, might simplify it further into RREF.

Augmented Matrix

An augmented matrix is an essential tool in linear algebra when representing a system of linear equations. It involves forming a matrix that consists of the coefficients of the variables from the equations and incorporates the constants from the equation results as an additional column.

Here's why it's used:

- **Simplification**: It converts systems of equations into a single matrix, making it more manageable to apply row operations.

- **Equation Representation**: The augmented matrix method allows us to visualize and perform calculations directly, without continuously referring to the equations in their original form.


For the given exercise, the matrix presented is the augmented matrix for a system of linear equations. Each row corresponds to an equation, and each column corresponds to a variable's coefficient, with the final column representing the constants in each equation.

System of Equations

A system of equations consists of multiple equations that share common variables. It represents conditions that the variables must satisfy simultaneously. In contexts such as this one, systems are often converted to forms that simplify finding solutions.

How to convert a matrix back into a system of equations:

• Identify each row of the matrix to correspond to one equation in the system.
• Align each column with a variable in the system, except for the rightmost column which represents the constant term on the equation's right side.

For example, using the exercise matrix, we transform it into the following system:

- First equation: From the first row you get \( x_1 + 3x_2 + x_4 = 0 \)
- Second equation: From the second row you get \( x_2 + 4x_4 = 0 \)
- Third equation: From the third row you get \( x_4 + x_5 = 2 \)
- Fourth equation: From the final row you get \( x_4 = 0 \)

This conversion is pivotal as it allows mathematical solutions to populate the variables effectively and efficiently with methodologies like substitution and elimination.