Problem 36 - Algebra for College Students

Question

In Problems 35-42, each matrix is the reduced echelon matrix for a system with variables \(x_{1}, x_{2}, x_{3}\), and \(x_{4}\). Find the solution set of each system. \(\left[\begin{array}{rrrr:r}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & -5 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 4\end{array}\right]\)

Step-by-Step Solution

Answer

The solution is a unique set: \(x_1 = 0, x_2 = -5, x_3 = 0, x_4 = 4\).

Step 1: Identify the Form of the Matrix

The given matrix is in reduced row echelon form, where each row represents an equation, and the last column represents the constants on the right side of the equations. The identity matrix in the first four columns indicates that each of the variables is leading (pivotal) in its equation.

Step 2: Write Down the Corresponding Equations

Based on the rows of the reduced echelon matrix, we can write down the system of equations for the variables: - From the first row: \(x_1 = 0\)- From the second row: \(x_2 = -5\)- From the third row: \(x_3 = 0\)- From the fourth row: \(x_4 = 4\).

Step 3: Describe the Solution Set

Since the matrix is in reduced row echelon form with each variable having a leading 1 and no free variables, the solution set consists of a unique solution. The solution is simply each variable equal to the constant found in its corresponding row; there are no parameters or free variables involved.

Key Concepts

Matrix Equations

Matrix equations offer a compact and organized way to represent and solve systems of linear equations. A matrix equation typically takes the form of \(AX = B\), where \(A\) is a matrix that contains coefficients, \(X\) is the column matrix of variables, and \(B\) is the column matrix of constants. In the context of the given problem, an augmented matrix is used, which combines \(A\) and \(B\) into one single matrix. This merged format is especially useful for applying matrix operations such as row reductions to find solutions more efficiently.
Each row in the matrix equation corresponds to an individual linear equation in the system. For instance, a row \([1, 0, 0, 0 : 0]\) represents the equation \(x_1 = 0\). This structured approach allows for analyzing multiple equations simultaneously and visually understanding the relationships between variables.
Using row reduction techniques, like transforming a matrix to reduced row echelon form (RREF), simplifies the matrix equation vastly, uncovering the values of variables directly. With an RREF, each leading coefficient is 1, and you'll find zeros elsewhere in the column. This indicates a clear solution path for each variable, ensuring a smooth transition from matrix form back to an easy-to-interpret system of equations.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. Solving such systems means finding values for each variable that satisfy all the given equations simultaneously. In our exercise, we observe a system represented by a matrix equation, highlighting a direct relationship between matrices and systems of equations.
The reduced row echelon matrix presented corresponds to this system:

• First equation: \(x_1 = 0\)
• Second equation: \(x_2 = -5\)
• Third equation: \(x_3 = 0\)
• Fourth equation: \(x_4 = 4\)

By converting the matrix into this set of linear equations, we gain clear insight into the solution without unnecessary complexity. The strategic conversion from matrix to equations streamlines problem-solving, focusing directly on each variable's relationship within the system. This forms a foundation for tackling both simple and complex real-world problems using algebraic methods.

Unique Solution Set

A solution set comprises the values that satisfy all equations within a system. In the given exercise, the matrix is in reduced row echelon form, leading us directly to a unique solution set. This means there is precisely one set of values for \(x_1, x_2, x_3,\) and \(x_4\) that satisfy all the equations.
Each variable has a leading 1 in its column, indicating it's a basic variable with a clear, defined value. There are no free variables or parameters present, which simplifies our solution set enormously:

• \(x_1 = 0\)
• \(x_2 = -5\)
• \(x_3 = 0\)
• \(x_4 = 4\)

These values constitute the unique solution set, pinpointing specific outcomes without ambiguity. This is critical in many practical applications, ensuring that computations lead to reliable and single, unambiguous results.
Understanding the implications of having a unique solution set is crucial for advancing further in mathematical problem-solving and real-life applications where clear-cut solutions are desired. It reflects the simplicity and elegance that mathematics is capable of delivering.