Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Indicate whether the matrix is in rowreduced form. \(\left[\begin{array}{ll|l}0 & 1 & 3 \\ 1 & 0 & 5\end{array}\right]\)

Step-by-Step Solution

Answer

Yes, the given matrix \(\left[\begin{array}{ll|l}0 & 1 & 3 \\\ 1 & 0 & 5\end{array}\right]\) is in row-reduced form, as it satisfies all the required conditions.

Step 1: Check for nonzero rows above rows of all zeros

There are no rows with all zeros, so this condition is satisfied.

Step 2: Check if the leading entry of each nonzero row is 1

In both rows, the leading entry is 1. So, this condition is also satisfied.

Step 3: Check if the pivot columns have zeros above and below the pivot

The first column has a pivot in the second row, and since there are no entries above it, this part is satisfied. The second column has a pivot in the first row, and there are no entries below it, so it also satisfies this condition. The given matrix satisfies all the required conditions to be in row-reduced form. So, we can now confidently state the result:

Step 4: Conclusion

Yes, the given matrix is in row-reduced form.

Key Concepts

Understanding Matrices

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are used in various scientific fields and computations. They serve as a compact way to represent and work with sets of linear equations or transformations. In our exercise, we're presented with a 2x3 matrix (two rows and three columns):

\[\left[\begin{array}{ll|l}0 & 1 & 3 \1 & 0 & 5\end{array}\right]\]
Understanding the structure of a matrix is crucial before diving into operations such as row reduction. Each element in a matrix can be referred to by its position (row number, column number). Matrices can also be manipulated through a variety of operations, including addition, multiplication, and row operations.

The Concept of a Pivot in Matrices

A pivot, within the context of matrices, refers to the first nonzero number in any row, reading from left to right. It plays a key role in various matrix operations, especially in procedures like Gaussian elimination and determining the row-reduced form of a matrix. In our matrix exercise example, we first look at the leading entry, known as the pivot, which is the number '1' in the second row, first column.

In matrix row operations, the pivot's position helps us to identify which rows can be added, swapped, or multiplied by scalars without altering the solution set of a system of equations. The eventual goal is to obtain a simplified form that's easier to work with, particularly for solving systems of linear equations.

Role of the Leading Entry

The leading entry in the context of a matrix, particularly when discussing row-reduced form, is always the pivot or the first nonzero number in a row. The leading entry is significant because it dictates the positions of the other elements in the row when using row operations to simplify the matrix.

The conditions for row-reduced form require each leading entry to be '1', and in our exercise, both rows meet this criterion. Moreover, each leading entry should be the only nonzero entry in its column to properly define the pivot columns. This is essential for solving systems and for matrix operations like determining inverses and ranks.

Importance of Nonzero Rows

Nonzero rows in a matrix are those that contain at least one nonzero element. They are important because they represent meaningful equations when a matrix is being used to solve a system of linear equations. When transforming a matrix to row-reduced form, it is important to sort the matrix so that any rows consisting entirely of zeros are at the bottom.

Identifying and dealing with nonzero rows is the first step in row reduction, as seen in our matrix exercise. Ensuring that each nonzero row starts with a '1' and is above any rows of all zeros is part of the process for achieving the row-reduced form. The presence or absence of nonzero rows can also affect the rank and solutions of the matrix.