Problem 50 - Precalculus

Question

Use row operations to transform each matrix to reduced row-echelon form. $$\left[\begin{array}{rrr|r} 2 & -1 & 0 & 1 \\ -1 & 0 & 1 & -2 \\ -2 & 1 & 0 & -1 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is in reduced row-echelon form: \[ \begin{bmatrix} 1 & 0 & -1 & 2 \\ 0 & 1 & -2 & 3 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} \]

Step 1: Formulate the Augmented Matrix

The problem provides us with a matrix that includes the coefficients and constants from a system of linear equations. The augmented matrix is already given in the problem:\[\begin{bmatrix}2 & -1 & 0 & 1 \-1 & 0 & 1 & -2 \-2 & 1 & 0 & -1 \\end{bmatrix}\]This matrix is ready for row operations to achieve reduced row-echelon form.

Step 2: Create a Leading 1 in the First Row

We'll start with the first pivot in the matrix. Divide the first row by 2, the first element in the top row, to make it a leading 1.\[\begin{bmatrix}1 & -\frac{1}{2} & 0 & \frac{1}{2} \-1 & 0 & 1 & -2 \-2 & 1 & 0 & -1 \\end{bmatrix}\]

Step 3: Zero out the First Column Below Pivot

Add the first row to the second row and add twice the first row to the third row to create zeros below the first pivot.\[\begin{bmatrix}1 & -\frac{1}{2} & 0 & \frac{1}{2} \0 & -\frac{1}{2} & 1 & -\frac{3}{2} \0 & 0 & 0 & 0 \\end{bmatrix}\]

Step 4: Create a Leading 1 in the Second Row

For the second pivot, divide the second row by \(-\frac{1}{2}\) to make the leading coefficient 1.\[\begin{bmatrix}1 & -\frac{1}{2} & 0 & \frac{1}{2} \0 & 1 & -2 & 3 \0 & 0 & 0 & 0 \\end{bmatrix}\]

Step 5: Zero out Non-zero Elements above Each Pivot

Add half of the second row to the first row to eliminate the non-zero elements above the pivot in the second column.\[\begin{bmatrix}1 & 0 & -1 & 2 \0 & 1 & -2 & 3 \0 & 0 & 0 & 0 \\end{bmatrix}\]The matrix is now in reduced row-echelon form.

Key Concepts

Row Operations

Row operations are a set of movements we perform on the rows of a matrix to manipulate it into a desired form, often to solve systems of linear equations. These row operations are crucial because they allow us to transform a matrix into its reduced row-echelon form, which helps in systematically solving equations.

There are three basic types of row operations:

• You can swap two rows.
• You can multiply a row by a non-zero scalar.
• You can add or subtract the multiple of one row from another row.

These operations are powerful because they do not change the solution set of the system of equations represented by the matrix. They are foundational in linear algebra, making complex problems more manageable.

Augmented Matrix

An augmented matrix is a representation of a system of linear equations. It includes not only the coefficients of the variables but also the constants on the right side of the equations. In other words, it's a shortcut from writing multiple equations to a single organized structure.

This matrix setup looks like:

• The coefficients of the variables are grouped on the left side
• A vertical line separates them from the constants on the right

For example, in the matrix given:\[\begin{bmatrix} 2 & -1 & 0 & 1 \-1 & 0 & 1 & -2 \-2 & 1 & 0 & -1 \\end{bmatrix}\]The part on the left represents the coefficients, and the column on the right after the vertical line is for the constants. This arrangement simplifies the process of applying row operations, especially when aiming to transform it into reduced row-echelon form.

Leading 1

The concept of a "leading 1" in a row of a matrix signifies a pivotal point that helps in forming the reduced row-echelon form. A leading 1 is the first non-zero number from the left in a row. Achieving a leading 1 is critical as it sets up the ability to zero out other numbers in the same column and solve for variables effectively.

To create a leading 1, you might divide the entire row by the number at the pivot position. For instance, if a row starts with a 2, as in the original example:\[\begin{bmatrix} 2 & -1 & 0 & 1 \-1 & 0 & 1 & -2 \-2 & 1 & 0 & -1 \\end{bmatrix}\]We divide the first row by 2 to make the leading element 1. This is a direct application of row manipulation, which prepares the matrix for further operations necessary to reach its reduced form.

Zeroing Out Columns

Zeroing out columns is a technique used in matrix transformation to simplify the structure. Once a leading 1 is established in a row, the next step often involves zeroing out all other entries in that column. This process helps isolate the leading 1 and clears a path for easier computations.

For example, after creating a leading 1, additional row operations are used to eliminate non-zero entries beneath it. In the matrix:\[\begin{bmatrix} 1 & -\frac{1}{2} & 0 & \frac{1}{2} \0 & -\frac{1}{2} & 1 & -\frac{3}{2} \0 & 0 & 0 & 0 \\end{bmatrix}\]We zero out the column entries below the first leading 1 by performing operations such as subtracting multiples of a row from another. This is essential for progressing the matrix towards a form where solutions to the represented equations are readily visible, specifically in the reduced row-echelon form.