Problem 14 - Algebra and Trigonometry

Question

7–14 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. $$\left[\begin{array}{cccccc}{1} & {3} & {0} & {1} & {0} & {0} \\ {0} & {1} & {0} & {4} & {0} & {0} \\ {0} & {0} & {0} & {1} & {1} & {2} \\ {0} & {0} & {0} & {1} & {0} & {0}\end{array}\right]$$

Step-by-Step Solution

Answer

(a) Yes, in row-echelon form; (b) No, not in reduced row-echelon form; (c) The system is: \(x_1 + 3x_2 + x_4 = 0\), \(x_2 + 4x_4 = 0\), \(x_4 + x_5 = 2\), \(x_4 = 0\).

Step 1: Define Row-Echelon Form

A matrix is in row-echelon form if: 1) Any rows consisting entirely of zeros are at the bottom, 2) The leading (first non-zero) entry of each non-zero row after the first occurs to the right of the leading entry of the previous row, 3) All entries in a column below a leading entry are zeros.

Step 2: Analyze the Matrix for Row-Echelon Form

The given matrix is: \[\begin{bmatrix}1 & 3 & 0 & 1 & 0 & 0 \0 & 1 & 0 & 4 & 0 & 0 \0 & 0 & 0 & 1 & 1 & 2 \0 & 0 & 0 & 1 & 0 & 0 \\end{bmatrix}\]1) No rows of entirely zeros are present. 2) Leading entries appear as follows: 1st row (column 1), 2nd row (column 2), 3rd and 4th rows (column 4). All leading entries satisfy the rightward shift rule from the row above. Thus, the matrix is in row-echelon form.

Step 3: Define Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if: 1) It is in row-echelon form, 2) The leading entry in each non-zero row is 1, and is the only non-zero entry in its column.

Step 4: Analyze the Matrix for Reduced Row-Echelon Form

To be in reduced row-echelon form, all columns with leading 1s should have zeros everywhere else. The given matrix has two nonzero entries in the 4th column. Therefore, it is not in reduced row-echelon form.

Step 5: Write the System of Equations

The matrix represents an augmented matrix for a system of equations (with 5 variables and one column of constants). It can be written as:\[\begin{align*}x_1 + 3x_2 + x_4 &= 0 \x_2 + 4x_4 &= 0 \x_4 + x_5 &= 2 \x_4 &= 0 \end{align*}\]

Key Concepts

Reduced Row Echelon Form

To understand the concept of Reduced Row Echelon Form (RREF), let's first consider what makes a matrix fit this description. A matrix is considered to be in RREF if it meets specific criteria:

• It must initially be in row-echelon form.
• The leading entry of each non-zero row is always 1.
• Each leading 1 is the only non-zero entry in its column.

These requirements make the RREF unique for any given matrix and particularly useful in solving systems of equations, as it shows clearly defined solutions. For the matrix in the exercise, we observe that while it is in row-echelon form, it fails to reach reduced row-echelon form because some columns with leading entries contain other non-zero elements, specifically in column 4, which has two leading entries. Every column with a leading 1 should only have zeros elsewhere.

Augmented Matrix

An augmented matrix is a compact way to express the system of linear equations. It incorporates both the coefficients of the variables and the constants from the equations into a single matrix form. This transformation is beneficial because it simplifies the process of using matrix operations to solve the system of equations.
In an augmented matrix, the last column usually represents the constants from the right-hand side of the equations.
In the given exercise, we are provided with a matrix:

• Rows represent individual equations.
• The columns, except the last one, represent the coefficients of each variable in the system.
• The final column shows the constants from each equation.

For example, if observed carefully, reading through horizontally from the given matrix yields the coefficients and constants forming the equations like so: \(x_1 + 3x_2 + x_4 = 0\). This matrix form makes it convenient to perform various row operations to solve the system efficiently.

System of Equations

A system of equations is simply a collection of two or more equations with the same set of unknowns. Solving a system involves finding the values of the unknowns that satisfy all the equations simultaneously.
In context, when given a matrix, each row can be translated to a linear equation.

• For instance, the first row of the matrix \([1, 3, 0, 1, 0, 0]\) translates to the equation \(x_1 + 3x_2 + x_4 = 0\).
• We transfer coefficients and constant terms directly from the matrix to form the linear equations.

Such representations allow us to apply matrix operations systematically, leading us to solutions effectively, be it via methods like Gaussian elimination or converting into augmented matrices to inspect rref.

Leading Entry

A leading entry in the matrix row context is the first non-zero number in a row of a matrix. It helps to define the position of leading coefficients when determining row-echelon and reduced row-echelon forms.

• The leading entry must always be located further to the right as you move down rows, creating a kind of staircase pattern if the matrix is in row-echelon form.
• In reduced row-echelon form, the leading entry should be 1 and the only non-zero entry in its column.

In our exercise example, each row of the matrix has a distinct leading entry:

• In Row 1, the leading entry is in column 1.
• In Row 2, it’s in column 2.
• Rows 3 and 4 both have their leading entries in column 4, which indicates it does not conform to a reduced row-echelon as there appear multiple leading entries in the same column.

Identifying these entries is crucial because they guide transformations that simplify a matrix into its canonical form, making it easier to understand and solve.