Problem 43 - Precalculus with Limits

Question

In Exercises 41-44, determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form. \( \left[\begin{array}{rr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 2 \\ \end{array}\right] \)

Step-by-Step Solution

Answer

The given matrix is in row-echelon form but not in reduced row-echelon form.

Step 1: Check row-echelon form

Look at all the rows. All rows are nonzero, so the condition of nonzero rows being above rows with all zeros is satisfied. For the leading coefficients: the first row has the leading coefficient in the first column, the second row has the leading coefficient in the second column which is to the right of the leading coefficient in the first row, and the third row has its leading coefficient in the fourth column, which again is to the right of the leading coefficient of the second row. Therefore, the given matrix is in row-echelon form.

Step 2: Check reduced row-echelon form

Now check if the given matrix is in reduced row-echelon form. As every column that has a leading 1 does not have all other entries in the respective column as zeros (for example, in the fourth column, the leading 1 in the third row has 1 and -1 in the same column), the given matrix is not in reduced row-echelon form.

Key Concepts

Matrix Transformations

Matrix transformations in linear algebra refer to the operations applied to matrices to achieve a specific form or solve a system of equations. Understanding matrix transformations is crucial because they allow us to simplify complex systems, making computations more straightforward and solutions easier to find.

Matrices can be transformed using several operations:

• Swapping two rows — this can help rearrange the matrix into a more useful form without changing the solution to a system of equations.
• Multiplying a row by a non-zero scalar — this ensures that the scale of a row does not interfere with computational operations.
• Adding or subtracting the multiple of one row to another — this operation helps in eliminating terms to simplify matrices.

Matrix transformations are fundamental steps in reaching either row-echelon form or reduced row-echelon form, both integral for solving linear systems effectively by hand or using computers.

Reduced Row-Echelon Form

Reduced row-echelon form (RREF) is a specific type of matrix form that simplifies solving systems of linear equations. A matrix is in RREF if it meets certain conditions, further refining the row-echelon form. In RREF:

• Each leading entry in every row is 1, known as a pivot, and these pivots are the only non-zero entries in their respective columns.
• The pivot 1 in any column should have zeros above and below it.
• All rows composed entirely of zeros are at the bottom of the matrix.
• The leading entry of each successive row is to the right of the leading entry of the row above.

Achieving RREF through matrix transformations provides a unique form that offers direct insights into the solutions of linear equations. It allows for straightforward backward substitution, leading to the simplest and most direct path to finding variable values.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It is essential for various disciplines including engineering, physics, computer science, and more.

Key elements of linear algebra involve matrices, vectors, determinants, and eigenvalues, among others. Matrices allow complex systems of equations to be compressed into a single matrix equation, making them easier to handle.

Central to linear algebra is the concept of solving systems of linear equations. This involves utilizing methods like Gaussian elimination, where matrices are transformed using elementary row operations to reach row-echelon form or RREF, helping to easily derive solutions.

Linear algebra serves as the mathematical underpinning of many geometric and physical problems, providing a framework for modeling real-world phenomena. Its operations and concepts are foundational for computational efficiency in algorithms utilized today in everything from data analysis to machine learning.