Problem 48 - Precalculus

Question

Use row operations to transform each matrix to reduced row-echelon form. $$\left[\begin{array}{ccc|c} -3 & -1 & 2 & -1 \\ -1 & -2 & 1 & -3 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix in reduced row-echelon form is: \( \begin{array}{ccc|c} 1 & 0 & -\frac{11}{15} & -\frac{1}{5} \\ 0 & 1 & -\frac{1}{5} & \frac{8}{5} \end{array} \).

Step 1: Make the First Pivot Element 1

The element in the first row, first column is -3. We need it to be 1 to begin creating the reduced row-echelon form. Divide the entire first row by -3:\[ \left[ \begin{array}{ccc|c} 1 & \frac{1}{3} & -\frac{2}{3} & \frac{1}{3} \ -1 & -2 & 1 & -3 \end{array} \right] \]

Step 2: Eliminate Below the First Pivot

To make the element below the first pivot (second row, first column) a 0, add the first row to the second row:\[ \left[ \begin{array}{ccc|c} 1 & \frac{1}{3} & -\frac{2}{3} & \frac{1}{3} \ 0 & -\frac{5}{3} & \frac{1}{3} & -\frac{8}{3} \end{array} \right] \]

Step 3: Make the Second Pivot Element 1

The current pivot element in the second row, second column is \(-\frac{5}{3}\). Divide the entire second row by \(-\frac{5}{3}\):\[ \left[ \begin{array}{ccc|c} 1 & \frac{1}{3} & -\frac{2}{3} & \frac{1}{3} \ 0 & 1 & -\frac{1}{5} & \frac{8}{5} \end{array} \right] \]

Step 4: Eliminate Above the Second Pivot

Make the second element of the first row (above the second pivot) zero by subtracting \(\frac{1}{3} \times\) the second row from the first row:\[ \left[ \begin{array}{ccc|c} 1 & 0 & -\frac{11}{15} & -\frac{1}{5} \ 0 & 1 & -\frac{1}{5} & \frac{8}{5} \end{array} \right] \]

Step 5: Clean Up the Matrix

We have fully eliminated above and below each pivot, and all pivots are 1. Perform any final simplifications if needed.Our final matrix in reduced row-echelon form is:\[ \left[ \begin{array}{ccc|c} 1 & 0 & -\frac{11}{15} & -\frac{1}{5} \ 0 & 1 & -\frac{1}{5} & \frac{8}{5} \end{array} \right] \]

Key Concepts

Row Operations

Row operations are fundamental techniques for manipulating matrices, especially when aiming to find their reduced row-echelon form (RREF). These operations allow you to transform a matrix while preserving the solutions to its corresponding system of linear equations. There are three primary types of row operations:

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

We use these operations because they simplify matrices in a systematic way, leading us to identify pivot elements and ultimately simplify the solution process for systems of equations. These row operations are applied sequentially to zero out elements below and above each pivot, hence efficiently reducing a matrix step by step.

Pivot Elements

Pivot elements serve as the anchor for transforming a matrix into its reduced row-echelon form. In a matrix, a pivot is typically a leading 1 in each row, used to eliminate other entries in its column. Think of pivots as markers that help guide the process of simplification through row operations. Here's how you can identify and use pivot elements:

• A pivot column must contain a leading 1.
• Every element below a pivot should be zeroed out using row operations.
• Once you set a pivot in a column, proceed to the next column where a new pivot is needed.

The presence of pivots not only helps organize the matrix but also indicates the rank of a matrix, helping us understand the underlying system's solution space.

Matrix Transformation

Matrix transformation is the process of altering a matrix from its initial form to a desired configuration, like the reduced row-echelon form. Throughout this process, various row operations are utilized to simplify and understand the given matrix better. Transformations typically involve:

• Converting the entries to create leading 1s, also known as pivots.
• Systematically applying row operations to remove other numbers, creating zeroes in strategic positions.

These transformations elucidate the solutions of systems by organizing the matrix into a form that is easy to interpret. The ultimate goal is to isolate variables, making it straightforward to solve for them.

Elimination Method

The elimination method simplifies systems of equations by focusing on removing variables, one per step, through strategic row operations. This method is intertwined with the row reduction process for matrices. Key elements in the elimination method include:

• Identifying pivot elements and using them to eliminate variables from other rows.
• Sequentially zeroing out terms below each pivot so that you isolate variables progressively.
• Efficiently transforming the matrix into its reduced row-echelon form.

By strategically eliminating variables, this method helps in resolving a system of equations into a simpler form, ultimately leading to an easier path to finding solutions.