Problem 10 - Precalculus: Mathematical for Calculus

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. $$\left[\begin{array}{rrrr} 1 & 0 & -7 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is in reduced row-echelon form and represents an inconsistent system.

Step 1: Identify Row-Echelon Form

A matrix is in row-echelon form if all non-zero rows are above any rows of all zeros, each leading coefficient (the first non-zero number from the left) of a row is to the right of the leading coefficient of the row above it. In the given matrix, the leading coefficient 1 in the first row is to the left of the leading coefficient 1 in the second row, and both these are above the row which has a leading coefficient placed further to the right. Furthermore, each leading entry (non-zero number) below a leading entry of a previous row is zero. Hence, this matrix is in row-echelon form.

Step 2: Check for Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if it is in row-echelon form and every leading coefficient is 1, and it is the only non-zero entry in its column. In the provided matrix, all leading coefficients are 1, and each of these leading entries is the only non-zero entry in its respective column. Therefore, this matrix is in reduced row-echelon form.

Step 3: Write the System of Equations

The given matrix is an augmented matrix representing a system of linear equations. Each row of the matrix corresponds to an equation. Extracting these, we have: 1. The first row gives: \(1x + 0y - 7z = 0\) which simplifies to \(x - 7z = 0\). 2. The second row gives: \(0x + 1y + 3z = 0\) which simplifies to \(y + 3z = 0\). 3. The third row gives: \(0x + 0y + 0z = 1\) which indicates an inconsistency as it represents the equation \(1 = 1\) which doesn't involve any variables. Thus, the system is inconsistent.

Key Concepts

Row-Echelon Form

The row-echelon form (REF) of a matrix is a simplified version that helps in solving linear equations systematically. A matrix is said to be in row-echelon form when it satisfies specific criteria:

• All zero rows, if any, appear at the bottom of the matrix.
• The leading coefficient (the first non-zero element from the left) in each non-zero row must appear strictly to the right of the leading coefficient in the preceding row.
• Beneath each leading coefficient, the entries in the column must be zero.

This form makes it easier to perform back-substitution to solve systems of equations. In the given example, the matrix:\[\begin{array}{rrrr}1 & 0 & -7 & 0 \0 & 1 & 3 & 0 \0 & 0 & 0 & 1 \\end{array}\]is in row-echelon form because the leading coefficients (1 in the first two rows) meet the above conditions, and the non-zero row is above the row of all zeros, satisfying the criteria.

Reduced Row-Echelon Form

The reduced row-echelon form (RREF) takes the simplification further than the row-echelon form. A matrix is in reduced row-echelon form if it meets the row-echelon form requirements, along with additional conditions:

• Every leading coefficient must be 1.
• The leading coefficient must be the only non-zero entry in its column.

Achieving this form allows for direct reading of the solutions to a system of linear equations, as each variable corresponds to exactly one leading 1. Looking at the matrix given:\[\begin{array}{rrrr}1 & 0 & -7 & 0 \0 & 1 & 3 & 0 \0 & 0 & 0 & 1 \\end{array}\]each leading 1 in the matrix is the only non-zero value in its column, verifying that this matrix is indeed in reduced row-echelon form. In practical problem-solving, RREF is invaluable for finding unique solutions or identifying inconsistencies, as shown by the third row, which denotes an impossible equation \(0 = 1\).

System of Linear Equations

A system of linear equations can be represented in matrix form, often as an augmented matrix that compactly displays the coefficients and constants of each equation. This method simplifies solving and provides insights into the system’s characteristics. The matrix provided:\[\begin{array}{rrrr}1 & 0 & -7 & 0 \0 & 1 & 3 & 0 \0 & 0 & 0 & 1 \\end{array}\]represents the following system of equations:

• The first row translates to the equation \(x - 7z = 0\).
• The second row becomes \(y + 3z = 0\).
• The third row, \(0 = 1\), indicates an inconsistency, suggesting that the system does not have a valid solution, as no assignment of values to \(x\), \(y\), and \(z\) can satisfy this equation.

Understanding the matrix representation allows one to quickly ascertain the nature of the solutions and see if a system is consistent, inconsistent, or dependent on arbitrary parameters.