Problem 51 - Precalculus with Limits

Question

In Exercises 49-54, use the matrix capabilities of a graphing utility to write the \(matrix\) in reduced row-echelon form. \( \left[\begin{array}{rrr} 1 & 2 & 3 & -5 \\ 1 & 2 & 4 & -9 \\ -2 & -4 & -4 & 3 \\ 4 & 8 & 11 & -14 \\ \end{array}\right] \)

Step-by-Step Solution

Answer

The matrix in reduced row-echelon form is \[\left[\begin{array}{rrr} 1 & 2 & 3 & -5 \ 0 & 0 & 1 & -4 \ 0 & 0 & 0 & -27 \ 0 & 0 & 0 & 0 \ \end{array}\right]\]

Step 1 - Adjust R1 and R2

To reduce the matrix, the goal is to create zeros in the lower triangle of the matrix. The first step is to attempt to make the entries below the first entry (1, in the first row and first column) to zero. Start by swapping R1 and R2. This will give \[\left[\begin{array}{rrr} 1 & 2 & 3 & -5 \ 1 & 2 & 4 & -9 \ -2 & -4 & -4 & 3 \ 4 & 8 & 11 & -14 \ \end{array}\right]\] Then, apply R3 = R3 + 2*R1 and R4 = R4 - 4*R1. The new matrix is \[\left[\begin{array}{rrr} 1 & 2 & 3 & -5 \ 0 & 0 & 1 & -4 \ 0 & 0 & -5 & -7 \ 0 & 0 & 1 & 6 \ \end{array}\right]

Step 2 - Adjust R3 and R4

Carry on with the goal of creating zeros in the lower triangle. Apply the operation R3 = R3 + 5*R2 and R4 = R4 - R2. The resulting matrix is \[\left[\begin{array}{rrr} 1 & 2 & 3 & -5 \ 0 & 0 & 1 & -4 \ 0 & 0 & 0 & -27 \ 0 & 0 & 0 & 10 \ \end{array}\right]

Step 3 - Create R4 = 0

Further steps to reduce the matrix can now be applied by making R4 zero. Apply the operation R4 = R4 + 0.37 * R3. The matrix becomes \[\left[\begin{array}{rrr} 1 & 2 & 3 & -5 \ 0 & 0 & 1 & -4 \ 0 & 0 & 0 & -27 \ 0 & 0 & 0 & 0 \ \end{array}\right]\] This is the reduced row-echelon form.

Key Concepts

Matrix Operations

Matrix operations, such as addition, subtraction, and multiplication, allow us to manipulate and solve matrix equations. These operations are the backbone of matrix transformations, making it possible to reach the reduced row-echelon form of a matrix. Using operations like row swapping, scalar multiplication, and row addition or subtraction, we systematically transform the matrix to solve linear equations.
For example, in the exercise, the operation R1 and R2 were swapped, and operations such as R3 = R3 + 2 * R1 were applied. These steps illustrate the versatility of matrix operations in modifying the matrix structure to achieve our goal. When manipulating matrices, always ensure each step adheres to the permissible operations in linear algebra, keeping track of each transformation to reach the desired form efficiently.

Graphing Utility

Graphing utilities can greatly enhance the efficiency of calculating matrix operations. They are tools available in graphing calculators and software that automate most complex arithmetic involved in matrix manipulation.
By employing graphing utilities, users can input matrices and execute operations like row reduction, obtaining results like the reduced row-echelon form swiftly. This can save time and reduce human error, particularly with larger matrices or more complicated operations. This automation allows students to focus on understanding the concepts and interpretations rather than getting bogged down in manual calculations. When using graphing utilities, ensure the correct input format for your matrix to avoid incorrect output.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations. By converting a given matrix into its reduced row-echelon form, the solutions to the system appear more clearly. This method involves a sequence of row operations: swapping rows, multiplying rows by nonzero scalars, and adding multiples of one row to another.
Implementing Gaussian elimination involves strategic operations to create zeros below each leading 1, known as pivoting. In our exercise, this is seen where R3 is adjusted to ensure zeros below the pivot in the first column. Repeatedly applying these steps moves us closer to a matrix that reveals the solutions to the system, or shows the dependencies of the system variables.

Linear Algebra

Linear algebra is the branch of mathematics focusing on vector spaces and linear mappings between these spaces. It is the foundation for many areas of mathematics and applied disciplines. Within linear algebra, matrix theory plays a crucial role in solving linear systems, where concepts such as the reduced row-echelon form help simplify and solve equations.
The reduced row-echelon form of a matrix indicates the structure and properties of the solutions. Linear algebra also delves into vector spaces' properties, helping us understand dimensions, kernel, rank, and more. Matrix transformations, like those seen in the exercise, illustrate linear algebra's power in making complex problems manageable, highlighting its applications in various fields from computer science to engineering.