Question

Write the matrix in reduced row-echelon form. $$ \left[\begin{array}{rrr} 1 & 3 & 2 \\ 5 & 15 & 9 \\ 2 & 6 & 10 \end{array}\right] $$

Short answer

The reduced row-echelon form of the given matrix is \[\left[\begin{array}{ccc}1 & 3 & 0 \0 & 0 & 1 \0 & 0 & 0\end{array}\right]\]

Step-by-step solution

Identifying the Pivot of the First Row

In a matrix, the pivot is the first non-zero element. For the given matrix, the pivot of the first row is 1 which is in the correct form, so no changes are needed for the first row.

Removing Multiples of the Pivot from the Other Rows

The goal is to make the elements beneath the pivot to be zero. To achieve this, subtract 5 times the first row from the second row and 2 times the first row from the third row. The new matrix is \[\left[\begin{array}{ccc}1 & 3 & 2 \0 & 0 & -1 \0 & 0 & 6\end{array}\right]\]

Identifying the Pivot of the Second Row

The pivot in the second row should ideally be in the second column, but it is 0. Thus, we proceed to the third row.

Swapping Rows to Get a Non-Zero Pivot in the Second Row

To get a non-zero pivot in the second row, swap the second and third rows. The new matrix becomes\[\left[\begin{array}{ccc}1 & 3 & 2 \0 & 0 & 6 \0 & 0 & -1\end{array}\right]\]

Dividing Second Row by the Pivot

Divide the second row by its pivot, which is 6. The new matrix is\[\left[\begin{array}{ccc}1 & 3 & 2 \0 & 0 & 1 \0 & 0 & -1\end{array}\right]\]

Removing Multiples of the Second Row Pivot from the Other Rows

Subtract the second row from the first and third rows. The final matrix in reduced row-echelon form is\[\left[\begin{array}{ccc}1 & 3 & 0 \0 & 0 & 1 \0 & 0 & 0\end{array}\right]\]

Key concepts

Understanding the Pivot

In the realm of matrices, a pivot is a central concept. It refers to the first non-zero element in a row of a matrix, usually used in the process of row reduction. These pivots are significant because they help determine the reduced row-echelon form (RREF) of a matrix. Finding the pivot is often the first step in transforming a matrix using row operations. For example, in our original matrix, the pivot in the first row is the number 1, because it is the first non-zero element. Once identified, we use this pivot to simplify the rest of the matrix. This involves ensuring all elements below this pivot become zeros, thus solidifying its position and simplifying the matrix. A good understanding of pivots is crucial for correctly applying further transformations and achieving the RREF.

Matrix Transformation Basics

When we talk about matrix transformation, we're essentially referring to the process of changing a matrix's structure while maintaining its core characteristics. The goal often is to simplify the matrix to an easily understandable form. The reduced row-echelon form is one such simplified form useful in solving systems of linear equations. When transforming a matrix, pivots play an instrumental role in guiding each transformation step. Initially, we locate the pivot and then use it to zero out all elements below it in its column. This is achieved through a series of row operations, which are basic transformations aimed at reaching RREF. After dealing with a pivot, you move to the next row to find its pivot, repeating the transformation process. This systematic approach ensures that each pivot is correctly placed, leading ultimately to a matrix expression that's straightforward and interpretable.

Elementary Row Operations

Elementary row operations are the tools we use to perform matrix transformations. There are three primary operations: row swapping, row multiplication by a nonzero scalar, and row addition. Each of these operations plays a distinct role in manipulating matrices towards RREF.

• Row Swapping: This operation involves exchanging two rows of a matrix. It can be crucial when a row with a pivot needs to be repositioned to help simplify the matrix, as we saw when the second row had to be swapped with the third row to get a non-zero pivot.
• Row Multiplication: Multiplying an entire row by a nonzero scalar can help in scaling pivots to 1, making it easier to manage operations in subsequent steps. For example, dividing the second row by 6 to turn the pivot into 1.
• Row Addition/Subtraction: By adding or subtracting rows (or multiples of rows), we can eliminate values and create zeros strategically. In our exercise, we subtracted multiples of the pivot rows from others, a common tactic to simplify and isolate pivots effectively.

Understanding these operations is vital for effective row reduction, and with practice, they become invaluable tools for solving complex matrix challenges.