Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Fill in the blank(s) to form a new row-equivalent matrix. Original Matrix New Row-Reduced Matrix .\(\left[\begin{array}{rrrr}1 & 5 & 4 & -1 \\ 0 & 1 & -2 & 2 \\ 0 & 0 & 1 & -7\end{array}\right] \quad\left[\begin{array}{rrrr}1 & 0 & & \\ 0 & 1 & -2 & 2 \\ 0 & 0 & 1 & -7\end{array}\right]\)

Step-by-Step Solution

Answer

The missing item on the new row-equivalent matrix is 14.

Step 1: Identify row operations

The matrix element that is changed is in the first row and second column. To make this change, we can subtract 5 times the second row from the first row.

Step 2: Substitute the Row Operation

To get the new first row, subtract five times each element in Row 2 from the corresponding element in Row 1. So, the new first row would be as follows: \(1 - 5 * 0\), \(5 - 5 * 1\), \(4 - 5 * -2\), \(-1 - 5 * 2\). This gives us \([1, 0, 14, -11]\). So the element that was missing in the new row-reduced matrix is 14.

Key Concepts

Matrix Row Operations

In the realm of linear algebra, matrix row operations are fundamental tools for manipulating matrices to simplify them or to achieve a particular form. Simply put, these operations can swap rows, multiply a row by a non-zero scalar, or add a multiple of one row to another. These are all permissible moves that maintain the matrix's essential properties, like the solutions to the system of linear equations it represents.

Let's look at how this applies to the given exercise. The original matrix was transformed into a new, row-equivalent matrix through a series of these operations. To zero out the entry in the first row and second column, one must perform a specific matrix row operation that involves subtracting a multiple of the second row from the first row.

• Subtraction of multiples: This operation is used when there is a need to create zeros in a row by using the non-zero entries from another row. By carefully selecting the right multiples, you can simplify the matrix, often aiming for an upper triangular form, which is a step towards the row-reduced echelon form.

Row Equivalence

The concept of row equivalence is central to understanding the relationship between various matrices. Two matrices are said to be row equivalent if one can be transformed into the other through a sequence of elementary row operations. This indicates that both matrices correspond to the same linear transformation or system of equations but are simply different representations.

In the context of the exercise, the original matrix and the new row-reduced matrix are row equivalent. Despite the second matrix looking different due to the zero in the first row, second column, it actually represents the same constraints as the original matrix. This ability to tweak the matrix without altering its inherent meaning is what makes row equivalence a powerful tool in solving linear systems and analyzing the matrix's properties, such as its rank.

• Preservation of Solutions: Importantly, when matrices are row equivalent, they have the same set of solutions to the associated linear system. Therefore, transforming a complex matrix into a simpler, but row-equivalent, form aids in finding these solutions with greater ease.

Elementary Row Operations

Digging deeper into matrix manipulation, we encounter the building blocks termed elementary row operations. There are three types that you will typically use:

• Type 1: Swap the positions of two rows. It's like saying, 'You take my place, and I'll take yours.'
• Type 2: Multiply a row by a non-zero constant, effectively scaling it up or down.
• Type 3: Add or subtract a multiple of one row to another row, a maneuver to introduce zeros or adjust values without altering the system.

Regarding the exercise at hand, an elementary operation of Type 3 was applied to zero out the value in the second column of the first row. Specifically, the second row was multiplied by 5—the coefficient of the second column in the first row—and then subtracted from the first row. This operation transforms the matrix to bring it closer to a row-reduced form, leading to an efficient way to solve the system of equations it represents.

Applying such operations can greatly simplify the process of finding solutions and revealing the underlying structure of the matrix. These techniques are essential for methods like Gaussian elimination, which streamlines solving linear systems.