Problem 8 - 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. $$\left[\begin{array}{rrr} 1 & 3 & -3 \\ 0 & 1 & 5 \end{array}\right]$$

Step-by-Step Solution

Answer

(a) Yes, row-echelon form. (b) No, not reduced row-echelon form. (c) System: \(x + 3y = -3\), \(y = 5\).

Step 1: Understand Row-Echelon Form

A matrix is in row-echelon form if it has following properties: 1. All zero rows, if any, are at the bottom of the matrix. 2. The leading entry of each nonzero row (after the first) is strictly to the right of the leading entry of the previous row. 3. All leading entries are 1.

Step 2: Analyze the Given Matrix for Row-Echelon Form

The matrix is \[ \begin{bmatrix} 1 & 3 & -3 \ 0 & 1 & 5 \end{bmatrix} \]It satisfies conditions for row-echelon form:- There are no zero rows.- The leading 1 of the second row (in column 2) is to the right of the leading 1 of the first row (in column 1).- All leading entries are 1. Therefore, it 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 the only nonzero entry in its column.

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

The matrix \[ \begin{bmatrix} 1 & 3 & -3 \ 0 & 1 & 5 \end{bmatrix} \]is not in reduced row-echelon form because:- In the first column, the leading 1 (row 1) is not the only non-zero entry; thus the entries above/below leading 1s in both columns should be zero.- Therefore, it is not in reduced row-echelon form.

Step 5: Write the System of Equations

The given matrix corresponds to a system of linear equations where each row represents an equation. Each column (except the last) represents a coefficient of variables, for example, \(x\) and \(y\), and the last column is the constant term. Thus, the system of equations is:1. \(1x + 3y = -3\)2. \(0x + 1y = 5\)So, the matrix represents the system of equations:- \(x + 3y = -3\)- \(y = 5\)

Key Concepts

Reduced Row Echelon Form

Reduced Row Echelon Form (RREF) is an advanced step in simplifying matrices in Linear Algebra. While a matrix in row-echelon form just requires that each leading coefficient is to the right of the one above, RREF adds more conditions to simplify it further. In RREF, each leading entry or 'pivot' in a row must be 1. Additionally, it mandates that the pivot must be the only nonzero entry in its column. This means all other entries in the column where a pivot is present must be zero.
This helps in making systems of linear equations clearer and often uniquely solvable, when looking at it from a practical perspective. RREF is particularly useful in solving systems of linear equations because it provides a direct path to the solution when expressed in its augmented form.

System of Linear Equations

A System of Linear Equations consists of multiple linear equations that share the same set of variables. Each equation represents a line or plane in space, and solutions to the system are points where all these lines or planes intersect. In matrix notation, such systems are often represented using augmented matrices, where each row corresponds to an equation.
For our given matrix, the system of equations becomes:

• \( x + 3y = -3 \)
• \( y = 5 \)

In the context of this system, we seek a solution that makes both these equations true simultaneously. This might involve finding values of \(x\) and \(y\) that satisfy both equations, thus representing the intersection point of the corresponding lines.

Matrix

A Matrix is a rectangular array of numbers arranged in rows and columns which is used extensively in Linear Algebra. Each element in a matrix can be part of mathematical computations, such as solving systems of equations. For example, our matrix is:ee\[ \begin{bmatrix} 1 & 3 & -3 \ 0 & 1 & 5 \end{bmatrix} \]each row can represent a linear equation's coefficients, aligning with variables such as \(x\), \(y\), etc.
In this context, matrices provide a powerful shorthand for handling multiple equations simultaneously, simplifying multi-variable problems into an elegant and manageable format. Additionally, matrices can be transformed through operations like addition, subtraction, multiplication, and transformations, aiding deeper insights and solutions.

Linear Algebra

Linear Algebra is a branch of mathematics dealing with vectors, vector spaces (also called linear spaces), and linear mappings between these spaces. It's a corner piece of scientific and engineering disciplines.
Linear Algebra involves studying systems of linear equations, and matrices play a pivotal role in it. By organizing these systems into matrices, Linear Algebra simplifies complex problems, making it easier to comprehend and solve them. It extends beyond solving equations, encompassing phenomena in computer graphics, engineering mechanics, data science, and much more.

• Linear transformations simplify geometric problems.
• Matrix decompositions help in signal processing.
• Eigenvectors and eigenvalues assist in understanding system dynamics.

Learning Linear Algebra equips you with a fundamental toolkit for exploring numerous practical and theoretical disciplines.