Problem 20 - Intermediate Algebra

Question

Perform each of the following elementary row operations on the augumented matrix \(\left[\begin{array}{rrr}-3 & 1 & -6 \\ 1 & -4 & 4\end{array}\right].\) $$ 3 R_{2}+R_{1} $$

Step-by-Step Solution

Answer

The matrix after the operation is: \[\left[\begin{array}{rrr}0 & -11 & 6 \\ 1 & -4 & 4\end{array}\right]\]

Step 1: Identifying the Row Operation

In the given exercise, you are required to perform the row operation \(3R_2 + R_1\) on the augmented matrix. This operation involves multiplying the second row (\(R_2\)) by 3, then adding the result to the first row (\(R_1\)).

Step 2: Multiplying the Second Row

Multiply each element of \(R_2 = [1, -4, 4]\) by 3. The resultant row after multiplication is: \[3R_2 = [3\times1, 3\times(-4), 3\times4] = [3, -12, 12]\]

Step 3: Adding the Rows

Add the resultant row \([3, -12, 12]\) to the first row \(R_1 = [-3, 1, -6]\).Perform the element-wise addition: \[\begin{array}{ccc}-3 & + & 3 & = & 0 \1 & + & (-12) & = & -11 \-6 & + & 12 & = & 6\end{array}\]The new first row is \([0, -11, 6]\).

Step 4: Updating the Matrix

Replace the original first row in the matrix with the new calculated first row \([0, -11, 6]\). Original second row remains unchanged.The updated matrix is: \[\left[\begin{array}{rrr}0 & -11 & 6 \ 1 & -4 & 4\end{array}\right]\]

Key Concepts

Augmented Matrix

An augmented matrix is a representation that allows us to handle a system of linear equations compactly by appending the columns corresponding to the "equals" part of equations to the matrix of their coefficients. For example, if you have equations like this:

• \( -3x + y = -6 \)
• \( x - 4y = 4 \)

You can form an augmented matrix by placing the coefficients of \( x \) and \( y \) in one matrix, and the constants on the right side of the equations as another column:\[\begin{bmatrix} -3 & 1 & -6 \ 1 & -4 & 4 \end{bmatrix}\]This simple structure helps in performing operations and finding solutions using techniques like Gaussian elimination. When operations are applied to the augmented matrix, it mirrors the transformations one would perform on the equations directly.

Matrix Row Transformations

Matrix row transformations, or elementary row operations, are techniques used to manipulate the rows of a matrix with the goal of solving systems of linear equations more easily. There are three types of operations you can perform:

• Row Switching: Swap two rows of the matrix.
• Row Multiplication: Multiply all entries of a row by a non-zero constant.
• Row Addition: Add or subtract the entries of one row to another.

Such transformations are crucial because they help convert the matrix to a more manageable form, such as row-echelon form or reduced row-echelon form. In the context of our exercise, the operation involved multiplying a row by a constant and adding the result to another row, demonstrating a careful application of row addition. This property can drastically simplify matrices, making it easier to derive solutions.

Linear Algebra

Linear algebra is an area of mathematics focusing on vector spaces and the linear mappings between these spaces. It involves studying concepts like vector spaces, linear transformations, matrices, and systems of linear equations. It is a fundamental toolkit for scientists and engineers for solving mathematical problems that can be represented in matrix form.
Linear algebra principles allow us to understand and work with multi-dimensional data, perform mathematical modeling and even computer graphics. This knowledge forms the basis for procedures like the Gaussian elimination method, which utilizes elementary row operations to solve systems of linear equations by simplifying them to the point where solutions can be directly found.
When faced with exercises involving matrices, such as the one in the given example, it is key to grasp how these operations interact with equations represented within matrix form to derive meaningful solutions in linear algebra.