Problem 3 - College Algebra with Corequisite Support

Question

Is there only one correct method of using row operations on a matrix? Try to explain two different row operations possible to solve the auqmented matrix $$ \left[\begin{array}{rr|r} 9 & 3 & 0 \\ 1 & -2 & 6 \end{array}\right] $$

Step-by-Step Solution

Answer

There are different correct methods for row operations on a matrix, including row swaps and row additions.

Step 1: Understanding Row Operations

Row operations are techniques used to manipulate the rows of a matrix in order to simplify it or solve a system of equations. For this exercise, our goal is to find different correct methods of using row operations on the given augmented matrix.

Step 2: Perform First Method - Row Swap

A common row operation is swapping rows to achieve a leading 1 in the first row. Here, swap Row 1 and Row 2 to achieve:\[\begin{bmatrix}1 & -2 & 6 \9 & 3 & 0\end{bmatrix}\]

Step 3: Perform Second Method - Row Addition

Another row operation involves adding or subtracting multiples of one row to another. Start with multiplying Row 1 by 9 and subtracting from Row 2 to make the first element of Row 2 zero:\(9R_1 - R_2 \rightarrow R_2\)This gives:\[\begin{bmatrix}9 & 3 & 0 \0 & -21 & 54\end{bmatrix}\]

Step 4: Analyze Both Methods

Both methods show valid approaches to reducing the matrix: the swap method simplifies the process of back substitution, while the addition method progresses toward an upper triangular form. Each approach seeks efficiency for solving systems but ultimately depends on context or preference.

Key Concepts

Augmented Matrix

An augmented matrix is a powerful tool for solving systems of equations. It combines the coefficients of the variables and the constants from the equations into a single matrix. This setup allows us to use matrix operations to manipulate and solve the system more efficiently.

For example, consider the system of equations:

• 9x + 3y = 0
• x - 2y = 6

This system can be rewritten in matrix form as \(\left[\begin{array}{rr|r} 9 & 3 & 0 \ 1 & -2 & 6 \end{array}\right]\).

The vertical line in the matrix separates the matrix into the coefficient matrix on the left and the constant matrix on the right. By manipulating this matrix using row operations, we can simplify and solve the system of equations.

Row Swap

Row swapping is a fundamental matrix operation used to rearrange the rows of an augmented matrix. The main goal is often to position a row with simpler coefficients, which can then be used as a pivot to simplify other rows.

In our example, performing a row swap allows us to have a leading 1 in the first row, which simplifies further operations. Swapping Row 1 and Row 2, we convert the matrix to\[\begin{bmatrix}1 & -2 & 6 \ 9 & 3 & 0\end{bmatrix}\].

This new arrangement may make the process of elimination or back substitution easier, especially if we are aiming for a row echelon form. Row swaps occur without changing the solution of the system of equations, making it a reliable method to start simplifying a matrix.

Row Addition

Row addition involves adding or subtracting multiples of one row to another in order to zero out certain elements in the matrix. This technique is an essential part of the Gauss-Jordan elimination process, which aims to systematically simplify a matrix.

In the context of our example, another option after row swapping is to work directly to create zeros below the leading entries. By multiplying Row 1 by 9 and subtracting this from Row 2, the matrix becomes:\[\begin{bmatrix}9 & 3 & 0 \ 0 & -21 & 54\end{bmatrix}\].

This step aims to eliminate the 9 in the first column of Row 2, creating a simpler, nearly upper-triangular matrix. Row addition is especially useful when targeting specific elements for elimination, making it one of the most versatile row operations.

System of Equations

A system of equations consists of multiple equations that are solved together, often sharing several unknown variables. Solutions to these systems exist where the equations intersect in the variable space.

The main goal when using matrices in this context is to reduce the complexity of solving these equations manually. By converting a system into an augmented matrix, we can perform row operations to primarily simplify calculations and capably tackle larger systems.

In solving our example system using our augmented matrix, we aim to isolate variables and solve for their values by appropriately transforming the matrix through row operations, ultimately reaching its row-echelon or reduced row-echelon form. This method is well-suited for systems with large numbers of equations and variables, where graphical methods or substitution might be cumbersome.