Problem 13 - Precalculus Mathematics 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. $$\left[\begin{array}{llll} 1 & 2 & 8 & 0 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

(a) Yes, in row-echelon form. (b) No, not in reduced row-echelon form. (c) Equations: \(x + 2y + 8z = 0\), \(y + 3z = 2\).

Step 1: Understand Row-Echelon Form

A matrix is in row-echelon form if each leading entry (the first nonzero number from the left, in a row) of every row is 1, and is also to the right of the leading entry of the row above it. There should also be any rows of all zeros at the bottom.

Step 2: Analyze the Matrix for Row-Echelon Form

The given matrix is \[\begin{array}{llll} 1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0 \end{array}\]. The leading entry of the first row is 1, the leading entry of the second row is to the right of the first row's, and the third row is all zeroes at the bottom. Hence, the matrix is in row-echelon form.

Step 3: Understand Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if it is in row-echelon form and each leading 1 is the only non-zero entry in its column, ensuring the columns containing leading 1s don't have any other non-zero entries.

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

Inspect the given matrix. The first column has a non-zero entry above a leading 1 which makes it not in reduced row-echelon form. Specifically, the element above the leading 1 in the second row is a 2.

Step 5: Write the System of Equations

Interpret the given augmented matrix into a system of linear equations. For the matrix \[\begin{array}{llll} 1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0 \end{array}\],the first row represents the equation \(1x + 2y + 8z = 0 \), and the second row represents \(1y + 3z = 2\). The third row, being all zeroes, does not contribute to the system.

Key Concepts

Row-Echelon Form

When working with matrices, understanding the row-echelon form is key. This form is specific; each row starts with a leading entry of 1, which is also the first non-zero number from the left in a row. Additionally, each leading entry must be positioned to the right of any leading entry in the row above it. Moreover, rows filled entirely with zeros should always be at the very bottom of the matrix.

In the matrix \[\begin{array}{cccc}1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0 \\end{array}\],we observe that the first row has a leading 1 in the first position and the second row's leading 1 is further right, fulfilling the positioning requirement. The matrix also appropriately places its row of zeros at the bottom, clearly meeting all the criteria for row-echelon form.

Reduced Row-Echelon Form

Reduced row-echelon form is a more refined version of row-echelon form. Not only must the matrix meet all the requirements of the row-echelon form, but the leading 1 in each row must be the sole non-zero entry in its column. This means that every column containing a leading 1 must have zeros in all other positions.

Looking at the example matrix, we see \[\begin{array}{cccc}1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0 \\end{array}\],we notice that, specifically in the first column, there is a 2 above the leading 1 in the second row. This breaks the rule of having zeros above or below the leading 1s, which prevents the matrix from being in reduced row-echelon form.

Augmented Matrix

An augmented matrix is a powerful tool for solving systems of equations. It involves writing out the coefficients and constant terms from a system of linear equations into a matrix, thus allowing for easier manipulation of the data. The convenience comes from using matrix operations such as Gaussian elimination to find the solutions without dealing directly with the equations.

Taking our specific matrix example\[\begin{array}{cccc}1 & 2 & 8 & 0 \0 & 1 & 3 & 2 \0 & 0 & 0 & 0 \\end{array}\],we construct the system of equations by recognizing each row as a separate equation based on the columns:

• First Row: \(1x + 2y + 8z = 0\)
• Second Row: \(1y + 3z = 2\)
• The third row doesn't contribute any equation since it's entirely zeros.

System of Equations

The relationship between an augmented matrix and a system of equations is foundational in solving linear algebra problems. A system of equations can have various outcomes based on the matrix - a unique solution, infinite solutions, or no solution - depending on how the rows interact and the resulting rank of the matrix.

For the matrix in question, we derive the system of equations

• \(x + 2y + 8z = 0\)
• \(y + 3z = 2\)

These equations can be solved using various methods, such as substitution or elimination, if needed. Understanding the matrix as an augmented matrix simply offers an organized method to visualize and tackle these systems efficiently. The presence of the third row of zeros indicates that there is no third equation, suggesting there might be dependent relationships or parameters when solving the system.