Problem 12 - 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. $$\left[\begin{array}{llll} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

(a) Yes, the matrix is in row-echelon form. (b) Yes, the matrix is in reduced row-echelon form. (c) The system is: \( x_1 = 1 \), \( x_2 = 2 \), \( x_3 = 3 \).

Step 1: Identify Row-Echelon Form Conditions

A matrix is in row-echelon form if it meets the following conditions: (1) All nonzero rows are above any rows of all zeros. (2) The leading entry of each nonzero 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. The given matrix shows these characteristics as its leading entries are 1, and each leading entry is to the right of the leading entry of the row above it, with zeros below each leading entry.

Step 2: Determine Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if, in addition to being in row-echelon form, each leading entry is 1 and is the only nonzero entry in its column. Reviewing the given matrix, each leading 1 is the only nonzero entry in its column, satisfying the additional condition for reduced row-echelon form.

Step 3: Construct System of Equations

In the augmented matrix, each row corresponds to an equation for the variables represented by the columns. The given matrix represents three equations. Assign the columns to variables as follows: the first three columns represent the coefficients of variables (\(x_1, x_2, x_3\)), and the last column represents the constants on the right. The system of equations is: \( x_1 = 1 \), \( x_2 = 2 \), and \( x_3 = 3 \).

Key Concepts

Reduced Row-Echelon Form

Understanding the concept of Reduced Row-Echelon Form (RREF) is crucial when working with matrices. A matrix is in RREF when it meets certain conditions that help in simplifying systems of equations:

• All nonzero rows appear above rows of all zeros.
• The first nonzero number from the left, also known as a leading 1, exists in each non-zero row, and it must appear to the right of the leading 1 in the above row.
• The leading 1 in each row must be the only nonzero number in its column.
• Any column containing a leading 1 has zeros in all its other entries.

RREF is particularly important because it helps us easily identify the solutions of a system of linear equations. For a given matrix like\[\begin{bmatrix}1 & 0 & 0 & 1 \0 & 1 & 0 & 2 \0 & 0 & 1 & 3 \end{bmatrix}\]the matrix satisfies all the RREF criteria. Each leading 1 is the only nonzero entry in its column. This allows us to determine the values of the variables directly from the matrix, making it a powerful tool for solving linear equations.

Augmented Matrix

An Augmented Matrix is a very handy representation used in linear algebra. It combines the coefficients of variables in a system of linear equations with the constants on the right side. Essentially, an augmented matrix merges the coefficient matrix and the constants matrix into one.
To create an augmented matrix, you place the constants on the right-hand side of an array that consists of the coefficients from the equations. For example, if we have the system of equations:

• \(x_1 = 1\)
• \(x_2 = 2\)
• \(x_3 = 3\)

The corresponding augmented matrix would be:\[\begin{bmatrix}1 & 0 & 0 & 1 \0 & 1 & 0 & 2 \0 & 0 & 1 & 3 \end{bmatrix}\]
This matrix seamlessly combines all elements required to define the system, simplifying analysis and computations, especially when attempting to solve the system using matrix operations. Analyzing an augmented matrix allows for the application of operations like Gauss-Jordan elimination, making it much easier to arrive at the solutions.

System of Equations

A System of Equations consists of multiple equations that are solved simultaneously. Each equation focuses on the relationship between various variables. The goal is to find the values of these variables that satisfy all equations in the system.
In linear algebra, a system can be represented through matrices, allowing for efficient computation. The process involves defining the relationships of the equations through matrices and then using techniques like Gaussian elimination to find solutions.
For example, using the given augmented matrix:\[\begin{bmatrix}1 & 0 & 0 & 1 \0 & 1 & 0 & 2 \0 & 0 & 1 & 3 \end{bmatrix}\]This matrix represents the system:

• \(x_1 = 1\)
• \(x_2 = 2\)
• \(x_3 = 3\)

These equations are immediately clear from the matrix, because each row corresponds directly to one equation, with the final column indicating the constants. Knowing how to translate between a system of equations and matrices enables one to solve complex problems more systematically and efficiently.