Problem 42 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. $$ \begin{aligned} &\text { I omitted row 3 from }\left[\begin{array}{rrr|r} {1} & {-1} & {-2} & {2} \\ {0} & {1} & {-10} & {-1} \\ {0} & {0} & {0} & {0} \end{array}\right] \text { and expressed }\\\ &\text { the system in the form }\left[\begin{array}{rrr|r} {1} & {-1} & {-2} & {2} \\ {0} & {1} & {-10} & {-1} \end{array}\right] \end{aligned} $$

Step-by-Step Solution

Answer

Yes, the statement makes sense because the removal of a row of zeros from a system of equations in matrix form does not change the set of solution.

Step 1: Understanding of Row Omission

The row to be omitted is the third row which is \(0, 0, 0 | 0\). This is called the zero row. Omission of a zero row from a system of equations or its matrix doesn't change the solution set. Therefore, omitting the third row is justifiable.

Step 2: Analysis of the Remaining System

Looking at the remaining system in the form \(\left[\begin{array}{rrr|r}{1} & {-1} & {-2} & {2} \ {0} & {1} & {-10} & {-1}\end{array}\right]\), it is clear that it represents the same set of equations as the original system but with the zero row omitted. Therefore, expression of the given system in this form is correct.

Key Concepts

Zero Row

In the context of matrix row operations, a zero row is a row in a matrix where all the elements are zero. This row effectively adds no information or equation to the system of equations that the matrix represents. - A zero row can often occur naturally during matrix manipulation, such as during Gaussian elimination, or when initially setting up the matrix. It simply means that the equation corresponding to that row results in a trivial identity, such as "0 = 0," offering no new insight about the solution set. - Because it has no effect on the relationships between variables in the system, removing a zero row does not alter the solution set. Think of a zero row as a placeholder—it exists in the matrix, but it doesn't tell us anything new about the system.

System of Equations

A system of equations is a collection of two or more equations with a common set of unknowns. - Systems of equations are often represented in matrix form, so solving them can be more straightforward. For example, the system given in the problem can be interpreted as: \[ \begin{align*} x - y - 2z &= 2 \ y - 10z &= -1 \end{align*} \]- Here, each row of the matrix corresponds to one linear equation in the system. The process involves writing down the coefficients of variables and the constant terms on one side—forming an augmented matrix. - By reformulating a system as an augmented matrix and using row operations, we can solve for the unknowns more effectively.

Solution Set

The solution set of a system of equations is the set of values that satisfy all equations in the system. - When considering an augmented matrix, the solution set consists of all possible combinations of variable assignments that make every equation in the system hold true. - In cases where the system of equations has infinite solutions, the solution set will contain infinite points, potentially forming a line or plane in geometric space. - The omission of a zero row doesn’t impact the solution set, because the zero row doesn't contribute any unique condition or constraint that needs to be satisfied.

Elementary Row Operations

Elementary row operations are fundamental tools used for simplifying matrices and solving systems of equations. These operations help manipulate the rows of matrices to derive solutions effectively. - There are three types of elementary row operations:

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

Using these operations strategically can transform a matrix into a more manageable form, such as row-echelon form or reduced row-echelon form, making it easier to extract the solution set. - For example, making the matrix "upper triangular" helps in back substitution, a method for finding solutions once the system is simplified. - These operations are the backbone of methods like Gaussian Elimination, which is invaluable in both theoretical and applied mathematics.