Problem 16 - Algebra and Trigonometry

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; (b) Yes; (c) \(x = 1\), \(y = 2\), \(z = 3\).

Step 1: Understand Row-Echelon Form

A matrix is in row-echelon form if: 1) All nonzero rows are above any rows of all zeroes. 2) Each leading entry of a row is in a column to the right of the leading entry of the preceding row. 3) All entries in a column below a leading entry are zero.

Step 2: Check Row-Echelon Conditions

Examine the matrix: \(\begin{bmatrix}1 & 0 & 0 & 1 \0 & 1 & 0 & 2 \0 & 0 & 1 & 3\end{bmatrix}\). Each leading 1 is to the right of the above row's leading 1, and all entries below leading ones are zero. Thus, the matrix is in row-echelon form.

Step 3: Understand 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 nonzero row is 1. 3) Each leading 1 is the only non-zero entry in its column.

Step 4: Check Reduced Row-Echelon Conditions

Review the matrix again: \(\begin{bmatrix}1 & 0 & 0 & 1 \0 & 1 & 0 & 2 \0 & 0 & 1 & 3\end{bmatrix}\). The leading entries are all 1s and are the only non-zero entries in their respective columns. The matrix is in reduced row-echelon form.

Step 5: Write Augmented Matrix as System of Equations

Convert the rows of the matrix to equations: \(1x + 0y + 0z = 1\), \(0x + 1y + 0z = 2\), and \(0x + 0y + 1z = 3\). The corresponding system of equations is: \(x = 1\), \(y = 2\), \(z = 3\).

Key Concepts

Row-Echelon Form

In the world of matrices, the row-echelon form serves as a foundational concept for simplification and solving linear equations. Understanding when a matrix is in row-echelon form is crucial, as this form allows us to easily interpret and work with the matrix.

A matrix meets the criteria for row-echelon form through the following rules:

• All non-zero rows must be above any rows comprised entirely of zeroes.
• Each leading entry, which is the first non-zero entry in a row, must appear to the right of the leading entry in any row directly above it.
• All entries below a leading entry within a column should be zero.

This structure systematically arranges the matrix and ensures ease of interpretation, paving the way for further processing like reaching the reduced row-echelon form.

Reduced Row-Echelon Form

The reduced row-echelon form is a form of a matrix that builds upon the row-echelon form, offering a more simplified structure. This form is particularly powerful because it provides a clear solution to systems of linear equations when used alongside augmented matrices.

Reduced row-echelon form adheres to the following stricter guidelines:

• The matrix must first be in the row-echelon form.
• The leading entry in each non-zero row must be the number 1.
• Each leading 1 must be the sole non-zero number in its respective column.

This format ensures that each variable's coefficient is clearly defined and isolated, making it significantly simpler to read off the solutions to a linear system from the final column of the augmented matrix.

Augmented Matrix

An augmented matrix is a mathematical construct that merges a system of linear equations into a single, convenient matrix form. It is designed by appending the constants from the equations' right-hand sides as an additional column to the right of the coefficient matrix.

This framework allows for the easy manipulation of the equations using matrix operations, facilitating algorithms to solve or simplify the system. Each row in an augmented matrix corresponds to a specific linear equation from the system, with columns representing the coefficients of the variables.

For instance, given the matrix \[ \begin{bmatrix} 1 & 0 & 0 & 1 \ 0 & 1 & 0 & 2 \ 0 & 0 & 1 & 3 \end{bmatrix} \]we can identify that the equations represented are \(x = 1\), \(y = 2\), and \(z = 3\). This makes the augmented matrix a powerful tool in linear algebra, especially when combined with row operations to bring matrices to row-echelon or reduced row-echelon form.

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. These equations represent different planes in the space determined by the number of variables. The solution to this system is the set of variable values that satisfy all the equations simultaneously.

When written in an augmented matrix form, these systems become easier to solve using matrix techniques, such as employing operations to reach row-echelon or reduced row-echelon form. This process allows mathematicians to transform complex multidimensional problems into manageable steps.

For example, the system of equations corresponding to the augmented matrix\[\begin{bmatrix} 1 & 0 & 0 & 1 \ 0 & 1 & 0 & 2 \ 0 & 0 & 1 & 3 \end{bmatrix} \]is \(x = 1\), \(y = 2\), and \(z = 3\). Each row represents an equation within our system, and the matrix form simplifies both the visualization and solution of these equations. This approach is particularly useful in linear algebra, engineering, and physics, where such systems frequently arise.