Problem 14 - 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. $$\left[\begin{array}{llllll} 1 & 3 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

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

Step 1: Check Leading Entries for Row-Echelon Form

Examine the matrix to determine if each leading entry (leftmost nonzero entry in a row) of a non-zero row is to the right of the leading entry of the row above. Additionally, each leading entry should be 1, and any zero rows (such as rows with all zeros) should be at the bottom. Given the matrix \(\left[\begin{array}{llllll}\ 1 & 3 & 0 & 1 & 0 & 0 \ 0 & 1 & 0 & 4 & 0 & 0 \ 0 & 0 & 0 & 1 & 1 & 2 \ 0 & 0 & 0 & 1 & 0 & 0 \ \end{array}\right]\),\ we see that the leading 1 in the second row is to the right of the leading 1 in the first row, and so on. This matrix is in row-echelon form.

Step 2: Check for Reduced Row-Echelon Form

For reduced row-echelon form, each leading 1 must be the only non-zero entry in its column. Check each column of the leading entries: the leading 1's in the first and second rows are already the only non-zero entries in their columns. However, the third and fourth rows both have leading 1's in the same column, which violates the requirements for reduced row-echelon form. Hence, the matrix is not in reduced row-echelon form.

Step 3: Write the System of Equations

To convert an augmented matrix back into a system of equations, treat each row as an equation. From the given matrix, we can write:1. \(x_1 + 3x_2 + x_4 = 0\)2. \(x_2 + 4x_4 = 0\)3. \(x_4 + x_5 = 2\)4. \(x_4 = 0\)where each variable corresponds to a column of the matrix, excluding the last which represents the constant terms.

Key Concepts

Augmented Matrix

An augmented matrix combines a coefficient matrix with an additional column to represent the constants from the equations. This single matrix form allows for efficient manipulation using matrix techniques. In the matrix provided, notice how the last column is separated from others.
This separation indicates the constants on the right side of the equations. By analyzing each row from the matrix, you can decipher individual equations.

• The structure helps to maintain proper alignment for applying matrix operations such as row interchange, scaling, and addition across the system of equations.
• This format simplifies solving linear equations by using systematic procedures like Gaussian elimination.

Recognizing an augmented matrix can streamline the process of transforming it back into a system of equations, ensuring each row directly translates to one equation.

Reduced Row-Echelon Form

The reduced row-echelon form (RREF) is a refined version of a row-echelon form. It's usually the desired final form when solving systems of linear equations because it clearly indicates the solution.
Unlike the row-echelon form, in RREF, the leading 1 of each row must be isolated with all other entries in its column being zero.
This gives you clear visibility of the solution.

• All nonzero rows are above rows of all zeros if any exist.
• The leading entry in each non-zero row is 1 and is the only non-zero entry in the column.

This matrix transformation ensures simplification of systems to the point where the solution can be read directly from the matrix. If a matrix fulfills these RREF conditions, solving becomes much more straightforward, making computation and interpretation seamless.

System of Equations

A system of equations is a collection of multiple equations that are solved together because they share variables. Each equation provides constraints that the values of the unknowns must satisfy simultaneously.
A system of equations can have various solution sets including no solution, exactly one solution, or infinitely many solutions.

• To express a system from an augmented matrix, focus on each row as a representation of a single equation.
• The number of equations typically corresponds to the number of rows in the matrix, while the number of unknowns corresponds to the columns minus the constant column.

For instance, converting rows of the given matrix yields equations in terms of unknowns like \(x_1, x_2\) and so on.
Understanding this connection between a matrix and its system of equations is crucial in analyzing and solving mathematical problems systematically.