Problem 2 - Algebra and Trigonometry

Question

Write the augmented matrix of the following system of equations. $$ \begin{aligned} \text { System } & \\\\\left\\{\begin{aligned} x+y-z=& 1 \\\ x+2 z=&-3 \\ 2 y-z=& 3 \end{aligned}\right.\end{aligned} $$ $$ \text { Augmented matrix } $$ Table cannot copy

Step-by-Step Solution

Answer

The augmented matrix is \([\begin{bmatrix} 1 & 1 & -1 & 1 \\ 1 & 0 & 2 & -3 \\ 0 & 2 & -1 & 3 \end{bmatrix}].\)

Step 1: Identify Coefficients and Constants

For each equation in the system, identify the coefficients of each variable and the constant term. For the first equation \(x + y - z = 1\), the coefficients are 1, 1, and -1, with the constant 1. For the second equation \(x + 2z = -3\), the coefficients are 1, 0, and 2, with the constant -3. For the third equation \(2y - z = 3\), the coefficients are 0, 2, and -1, with the constant 3.

Step 2: Construct the Matrix

Write the coefficients and constants in matrix form. Arrange the coefficients of \(x\), \(y\), and \(z\) from each equation into rows of a matrix, appending the constant from the equations as the last column. This results in the matrix:\[\begin{bmatrix}1 & 1 & -1 & | & 1 \1 & 0 & 2 & | & -3 \0 & 2 & -1 & | & 3\end{bmatrix}\]

Step 3: Remove Augmented Line

Since an augmented matrix is typically written without the vertical separator, write the matrix without the '|' to give a clean presentation:\[\begin{bmatrix}1 & 1 & -1 & 1 \1 & 0 & 2 & -3 \0 & 2 & -1 & 3\end{bmatrix}\]

Key Concepts

Systems of Equations

A system of equations involves multiple equations that share common variables. These equations are solved together since changing one variable affects the others. For instance, consider the system below which has three linear equations with variables \(x\), \(y\), and \(z\): \[\begin{aligned}x + y - z =& 1 \x + 2z =& -3 \2y - z =& 3\end{aligned}\] Each equation in this system is a linear equation, meaning it graphically represents a plane in a three-dimensional space. The solution to this system is the point where all these planes intersect in space. This intersection point gives the values of \(x\), \(y\), and \(z\) that satisfy all equations simultaneously. Solving systems of equations can be performed using various methods, such as substitution, elimination, or matrix methods which we'll explore below. Understanding the interaction between the different equations and their variables is key when dealing with systems of equations.

Matrix Representation

Representing a system of equations as an augmented matrix streamlines the process of finding solutions. This representation captures the essential information of the system into a more compact and manipulable form. Let's consider the given system:- \(x + y - z = 1\)- \(x + 2z = -3\)- \(2y - z = 3\) The coefficients of the variables \(x\), \(y\), and \(z\) are extracted and aligned in columns, while their corresponding constants are placed in a separate column. This forms the augmented matrix as follows: \[\begin{bmatrix}1 & 1 & -1 & | & 1 \1 & 0 & 2 & | & -3 \0 & 2 & -1 & | & 3\end{bmatrix}\] The vertical line is often depicted to segregate the constants from the coefficients, but in practice, we omit it for a clearer matrix presentation: \[\begin{bmatrix}1 & 1 & -1 & 1 \1 & 0 & 2 & -3 \0 & 2 & -1 & 3\end{bmatrix}\] This matrix succinctly encapsulates the system, allowing us to manipulate it through row operations to eventually find solutions.

Linear Algebra

Linear algebra is a branch of mathematics that centers around the study of vectors, vector spaces, and linear equations. It provides a universal language to represent and solve systems of equations efficiently through matrices. Matrix algebra is an integral part of linear algebra. Matrices can represent linear systems, transformations, and more. In solving systems of linear equations, operations such as row reduction can lead to an easier form known as the row-echelon form or reduced row-echelon form. These forms simplify finding solutions to the systems. In the context of our example, the linear algebra principles make it possible to apply algorithmic strategies such as Gaussian elimination or matrix inversion to find solutions. By representing equations as matrices, linear algebra reduces complex problems into manageable calculations. Moreover, the concepts of rank, determinants, and invertibility further complement the understanding of when systems have unique solutions, no solution, or infinitely many solutions. Linear algebra thus expands beyond mere computation; it offers a deeper insight into the nature and characteristics of the solutions encountered in systems of equations.