Problem 22 - College Algebra

Question

In Exercises \(19-24\), perform each matrix row operation and write the new matrix. \(\left[\begin{array}{rrr|r}1 & -3 & 2 & 0 \\ 3 & 1 & -1 & 7 \\ 2 & -2 & 1 & 3\end{array}\right] \quad-3 R_{1}+R_{2}\)

Step-by-Step Solution

Answer

The new matrix after the operation \(-3R_{1} + R_{2}\) is \(\left[\begin{array}{rrr|r} 1 & -3 & 2 & 0 \ 0 & 10 & -7 & 7 \ 2 & -2 & 1 & 3\end{array}\right]\).

Step 1: Identify the Rows Involved

The operation \(-3R_{1} + R_{2}\) indicates we will be manipulating rows 1 and 2, performing arithmetic operations on their elements.

Step 2: Perform the Operation

Multiply each element in row 1 by -3. The result is \([-3, 9, -6, 0]\). Then, add this to the corresponding elements in row 2. With the given row 2 as \([3, 1, -1, 7]\), this operation gives a new row 2: \([0, 10, -7, 7]\).

Step 3: Write the New Matrix

With the new row 2, write the new matrix with the original row 1 and row 3, bending the new row 2. The resulting matrix is \(\left[\begin{array}{rrr|r} 1 & -3 & 2 & 0 \ 0 & 10 & -7 & 7 \ 2 & -2 & 1 & 3\end{array}\right]\).

Key Concepts

Elementary Row Operations

Elementary row operations are the basic operations you can perform on the rows of a matrix. These operations are essential in topics like solving systems of linear equations and finding the inverse of a matrix. There are three main types of operations:

• Swapping two rows: This changes the position of rows but keeps the values intact.
• Multiplying a row by a nonzero constant: This scales the values of all elements in a row by the given constant.
• Adding or subtracting a multiple of one row to another row: This operation is useful for eliminating variables or simplifying equations.

In the given exercise, we focus on the third operation, where we multiply the first row by elements by -3 and then add them to the corresponding elements in the second row. Remember, these operations do not change the solution set of a system of linear equations but rather transform it into a form that's easier to work with.

Augmented Matrix

An augmented matrix is a convenient way to represent a system of linear equations. It combines the coefficient matrix of the variables with the constants from the equations into one single matrix. This structure is especially useful for performing row operations and applying techniques such as Gaussian elimination.
In an augmented matrix, a vertical bar is used to separate the column of constants from the coefficients of variables. In our exercise, the augmented matrix is represented as:\[\begin{bmatrix}1 & -3 & 2 & | & 0 \3 & 1 & -1 & | & 7 \2 & -2 & 1 & | & 3\end{bmatrix}\]
This matrix includes variables on one side and constants on the other, making it clear to see the relationship between the equations. It's a go-to format for systematic solution strategies involving matrix operations.

Matrix Manipulation

Matrix manipulation involves altering the rows of a matrix through various operations. This manipulation is instrumental in both simplifying matrices and solving them using specific techniques. In the context of our exercise, matrix manipulation helps in transforming the matrix into a new form where it's easier to identify solutions.
Operations like multiplying rows, swapping them, or adding multiples of rows help in creating zeros in strategic positions, also known as row-echelon form. This is crucial when solving linear systems, making matrix manipulation a powerful tool.
By employing these strategies, one can simplify complex systems, often converting them into forms where back-substitution can be easily applied to reach the final solution.

Matrix Transformation

Matrix transformation refers to modifying a matrix by changing the rows through calculations and operations. Each of these transformations maintains the equivalence of the matrix in terms of the solutions they represent. They do not alter the actual solution set, only the form of the matrix.
The exercise uses matrix transformation to simplify the problem. By performing operations on the augmented matrix, you modify its structure but not its essence — the system of equations still holds true.
Such transformations are tactical, often aiming to bring matrices into row-reduced forms that are much easier to solve through methods like forward elimination and back substitution. These transformations are at the heart of linear algebra techniques used in computation and theoretical understanding.