Problem 30 - Precalculus

Question

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form. $$\left[\begin{array}{llll|l} 1 & 0 & 0 & 1 & 3 \\ 0 & 1 & 0 & 3 & 2 \\ 0 & 0 & 1 & 0 & 5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is in reduced row-echelon form.

Step 1: Identify Row-Echelon Form

A matrix is in row-echelon form if it satisfies the following conditions: (a) All nonzero rows are above rows of all zeros; (b) Each leading entry of a row is in a column to the right of the leading entry of the row above it; and (c) All entries in a column below a leading entry are zero. In the given matrix, there is one row of zeros at the bottom, and each subsequent nonzero row has its leading entry further right. Thus, this matrix is in row-echelon form.

Step 2: Identify Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if it meets the criteria for row-echelon form and additionally satisfies: (d) The leading entry in each nonzero row is 1; and (e) Each leading 1 is the only nonzero entry in its column. In the given matrix, the leading entries are all 1 (the pivots in each nonzero row), and each of these 1s is the only nonzero entry in their respective columns. Thus, this matrix is in reduced row-echelon form as well.

Key Concepts

Reduced Row-Echelon Form

Matrices play a significant role in many fields, including linear algebra, and finding the reduced row-echelon form (RREF) of a matrix is a foundational skill. Reduced row-echelon form is a special type of matrix representing how one can simplify a system of linear equations. To consider a matrix in this form, certain conditions must be met:

• It has to satisfy all the conditions of being in row-echelon form, meaning that each nonzero row has a leading 1, with all zero rows at the bottom.
• The leading 1 in each nonzero row is to the right of any leading 1s in previous rows, creating a stair-step pattern.
• Each leading 1 must be the only nonzero entry in its column.

A matrix in reduced row-echelon form often simplifies solving systems of linear equations as these forms make it much easier to identify solutions directly. It's akin to breaking down complex problems into their simplest parts, revealing the core structure without any extraneous elements obscuring the view.

Matrix Algebra

Matrix algebra involves a variety of operations and transformations that are essential for understanding and solving linear systems. These operations include matrix addition, subtraction, scalar multiplication, and matrix multiplication. Each of these can manipulate matrices to form new matrices, often revealing insights into the systems they represent.

When working with matrices, turning them into simpler forms like row-echelon form or reduced row-echelon form enables easier manipulation and interpretation.

You might locate matrices in various applications, from computer graphics to modeling real-world phenomena in engineering. Understanding how to apply matrix algebra to reduce matrices to simpler forms helps in simplifying complex linear systems, and is a powerful tool in the mathematician's toolkit.

Linear Algebra Concepts

At the heart of linear algebra lie fundamental concepts like vectors, matrices, determinants, and eigenvalues. These elements are crucial for solving systems of linear equations and understanding linear transformations.

• Vectors: They are essentially numbers indicating position, direction, and magnitude in a space.
• Matrices: As an extension of vectors, matrices are arrays of numbers organized in rows and columns to represent linear transformations.
• Determinants: These are values associated with a square matrix that provide insight into the properties of the matrix, such as whether it is invertible.

Linear algebra's beauty lies in its ability to simplify problems involving linear equations and transformations, making it integral to both theoretical mathematics and applied fields. The study and application of these fundamental concepts enable individuals to unlock solutions that seem complex on the surface but are quite structured underneath.