Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Write the matrix in reduced row-echelon form. $$ \left[\begin{array}{rrrr} 1 & 2 & 3 & -5 \\ 1 & 2 & 4 & -9 \\ -2 & -4 & -4 & 3 \\ 4 & 8 & 11 & -14 \end{array}\right] $$

Step-by-Step Solution

Answer

The given matrix in reduced row-echelon form is \( \left[\begin{array}{rrrr} 1 & 2 & 0 & 1 \ 0 & 0 & 1 & -4 \ 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 1 \end{array}\right] \).

Step 1: Setup

Start with the original matrix: \( \left[\begin{array}{rrrr} 1 & 2 & 3 & -5 \ 1 & 2 & 4 & -9 \ -2 & -4 & -4 & 3 \ 4 & 8 & 11 & -14 \end{array}\right] \).

Step 2: Row operations

Perform row operations to reach echelon form. Subtract row 1 from row 2 and add 2*row1 to row 3 and subtract 4*row1 from row 4 to obtain: \( \left[\begin{array}{rrrr} 1 & 2 & 3 & -5 \ 0 & 0 & 1 & -4 \ 0 & 0 & 2 & -7 \ 0 & 0 & -1 & 6 \end{array}\right] \).

Step 3: More row Operations

Continue with operations to get to the reduced row echelon form. Subtract row 2 from row 3 and add row 2 to row 4 to obtain: \( \left[\begin{array}{rrrr} 1 & 2 & 0 & 7 \ 0 & 0 & 1 & -4 \ 0 & 0 & 0 & 3 \ 0 & 0 & 0 & 2 \end{array}\right] \).

Step 4: Reach the Reduced Echelon Form

Perform a scalar multiplication in the last row (row 3 and row 4) by multiplied by 1/3 and 1/2 respectively and then subtract 2 times row 3 from row 1 to get the final reduced row echelon form: \( \left[\begin{array}{rrrr} 1 & 2 & 0 & 1 \ 0 & 0 & 1 & -4 \ 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 1 \end{array}\right] \).

Key Concepts

Matrix Operations

Matrix operations are essential tools in linear algebra, allowing you to manipulate matrices to solve systems of linear equations, among other applications. Operations on matrices include addition, subtraction, scalar multiplication, and multiplication between matrices. Each operation follows specific rules to ensure proper implementation.

To add or subtract matrices, they must share the same dimensions. You add or subtract corresponding elements. Scalar multiplication involves multiplying each element in the matrix by a scalar (a real number).

Matrix multiplication, however, is more complex. It requires the number of columns in the first matrix to equal the number of rows in the second matrix. The resulting matrix's element is calculated by multiplying the row of the first matrix by the column of the second matrix and summing the results. These operations combine to help perform more complex procedures like row reduction, which is crucial in finding the reduced row-echelon form.

Row Reduction

Row reduction, also known as Gaussian elimination, is a process used to simplify a matrix while maintaining the solutions to the equations it represents. The goal is to achieve a form known as row-echelon form and, for further simplification, the reduced row-echelon form.

The process involves several row operations:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting a multiple of one row to another row.

These operations help in creating zeros below leading coefficients (the first non-zero number in a row when moving left to right), aligning them in a "staircase" pattern. This structure simplifies solving equations, allowing you to easily back-substitute to find variable values.

For the given exercise, row reduction simplified the matrix to a form where each leading term in a row is 1 and is the only non-zero number in its column, known as reduced row-echelon form. This makes it much easier to solve complex linear equations.

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. It provides powerful techniques for solving problems in engineering, physics, computer science, and more.

A core component of linear algebra is the study of matrices, which serve as a framework for expressing complex equations. Matrices help to standardize and solve systems of linear equations using concepts like row reduction.

In linear algebra, reducing a matrix to its reduced row-echelon form is crucial. This form simplifies the matrix and preserves the solutions to the system it represents. By converting matrices into this form, you reveal a straightforward result, making it easier to solve. Linear algebra's applications extend to network theory, computer graphics, optimization problems, and many other fields, illustrating its deep utility and importance.