Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Given that the augmented matrix in row-reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist. \(\left[\begin{array}{rrr|r}1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 2\end{array}\right]\)

Step-by-Step Solution

Answer

The system of linear equations has a unique solution given by \(x_1 = 3\), \(x_2 = -1\), and \(x_3 = 2\).

Step 1: Find the number of solutions

Based on the matrix, there are no rows with the form [0 0 ... 0 | a] where a is non-zero. This implies that the system has at least one solution since there are no contradictions.

Step 2: Identify the solution(s)

From the row-reduced matrix, we can see that the variables are already isolated, and the system of equations is as follows: \(x_1 = 3\) \(x_2 = -1\) \(x_3 = 2\) So, the unique solution to the system is \(x_1 = 3\), \(x_2 = -1\), and \(x_3 = 2\).

Key Concepts

Augmented Matrix

An augmented matrix is a useful tool in linear algebra for solving systems of linear equations. It combines the coefficients of the variables and the constants from the equations into a single matrix. This compact form allows for easy application of matrix operations.
An augmented matrix is structured with vertical bars separating the coefficient matrix from the constants. For example, consider a system of equations:

• Equation 1: \(a_1x + b_1y = c_1\)
• Equation 2: \(a_2x + b_2y = c_2\)

The augmented matrix representation would be:\[\begin{array}{cc|c}a_1 & b_1 & c_1 \a_2 & b_2 & c_2 \\end{array}\]This representation is especially useful when solving systems using methods such as Gaussian elimination or Gauss-Jordan elimination.

Row-Reduced Form

Row-reduced form is a way to simplify a matrix to make solving linear systems easy. We apply operations such as row swapping, scaling rows, and adding multiples of one row to another. These operations help transform our matrix into a form where solutions to the corresponding system are clear.
In row-reduced form, the leading coefficient in each row is 1, and all elements above and below these leading ones are zero. Additionally, it makes the system very straightforward to write as equations. For instance, consider the row-reduced form:\[\begin{array}{cc|c}1 & 0 & 4 \0 & 1 & 3 \\end{array}\]This directly corresponds to a system of equations like:

• \(x = 4\)
• \(y = 3\)

Both Gaussian elimination and Gauss-Jordan elimination are effective methods in transforming any matrix into this simplified form.

System of Equations

A system of equations consists of multiple equations that are solved simultaneously. Each equation in the system can represent a line, plane, or another geometric entity in a multidimensional space.
There can be three possible scenarios when dealing with systems of linear equations:

• There is exactly one solution: The lines or planes intersect at a single point.
• There are infinitely many solutions: The lines or planes are the same, leading to overlapping solutions.
• There is no solution: The lines or planes are parallel and never meet.

In linear algebra, one of the main goals is to determine which of these scenarios applies to a given system. Tools such as matrix row operations can help identify the number of solutions by simplifying the system to a more understandable form.

Solution of a System

Identifying the solution of a system means finding the variable values that satisfy all equations simultaneously. For a unique solution, the row-reduced matrix will show a clear relationship with each variable straightforwardly defined.
If considering the given problem's augmented matrix, which is:\[\begin{array}{ccc|c}1 & 0 & 0 & 3 \0 & 1 & 0 & -1 \0 & 0 & 1 & 2 \\end{array}\]We see that it directly translates into the system:

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

When a system is given in row-reduced form like this, identifying solutions is simple. Each row clearly isolates a variable, making it easy to determine the exact values that solve all equations in the system.