Question

Use Gaussian Elimination to put the given matrix into reduced row echelon form. $$\left[\begin{array}{ccc}-2 & -4 & -8 \\ -2 & -3 & -5 \\ 2 & 3 & 6\end{array}\right]$$

Short answer

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

Step-by-step solution

Identify Leading Entry

The leading entry of the first row is currently \(-2\). To start the row operations, we first want it to be 1. To do this, we multiply the entire first row by \(-1/2\).

Scale First Row

Multiply the first row by \(-1/2\): \[ \begin{bmatrix} 1 & 2 & 4 \ -2 & -3 & -5 \ 2 & 3 & 6 \end{bmatrix} \]

Eliminate First Column Below Pivot

We need to make the elements below the first pivot (which is 1) in column 1, zero. For this, we:1. Add 2 times the first row to the second row.2. Subtract 2 times the first row from the third row.The new matrix looks like: \[ \begin{bmatrix} 1 & 2 & 4 \ 0 & 1 & 3 \ 0 & -1 & -2 \end{bmatrix} \]

Eliminate Second Column Below Pivot

We focus next on the second pivot, which is the second row's leading 1. Add the second row to the third row to eliminate the \(-1\) below the leading 1 in column 2. The updated matrix is:\[ \begin{bmatrix} 1 & 2 & 4 \ 0 & 1 & 3 \ 0 & 0 & 1 \end{bmatrix} \]

Eliminate Above Pivots

To achieve reduced row echelon form, eliminate the elements above the pivots. For this:1. Subtract 2 times the third row from the first row.2. Subtract 3 times the second row from the first row.The result:\[ \begin{bmatrix} 1 & 0 & -2 \ 0 & 1 & 3 \ 0 & 0 & 1 \end{bmatrix} \]

Final Matrix Verification

Verify that each pivot is 1 and every element above and below each pivot is 0. The matrix is now in reduced row echelon form.

Key concepts

reduced row echelon form

The reduced row echelon form (RREF) of a matrix is a specific configuration used in linear algebra to simplify systems of linear equations. A matrix in RREF has several defining characteristics:

• Each pivot is 1 and is the only non-zero entry in its column.
• Every pivot must be to the right of any pivots in the rows above it.
• All rows consisting entirely of zeros are at the bottom of the matrix.

The process to reach this form often involves various row operations that transform the matrix step by step. It's a critical tool in solving linear systems because it provides a straightforward way to read solutions from the matrix. With Gaussian Elimination, we use a series of row operations to systematically zero out entries below and above the pivot positions. This enables the easy finding of solutions when working with the corresponding linear equations.

matrix operations

Matrix operations are the building blocks for manipulating matrices in linear algebra. Essential operations include:

• Row Addition or Subtraction: Adding or subtracting one row from another.
• Row Scaling: Multiplying every element of a row by a non-zero scalar.
• Row Interchanging: Swapping two rows within the matrix.

These operations are pivotal in methods like Gaussian Elimination, where we use them to simplify matrices or solve systems of equations. Importantly, these operations do not change the solution set of the system represented by the matrix, making them safe transformations that maintain the system's integrity. The ability to perform these operations allows us to manipulate the matrix into a more convenient form, such as the RREF, which is easier to interpret.

linear algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces, linear transformations, and matrices. It's foundational to fields like physics, engineering, computer science, and more. Key concepts in linear algebra include:

• Vector spaces: Sets where vectors can be added together and scalar multiplied.
• Linear transformations: Functions that preserve vector addition and scalar multiplication.
• Matrices: Representations of linear transformations, allowing for computation and analysis.

By organizing and simplifying complex data, linear algebra provides crucial tools and insights necessary for problem-solving in various scientific and engineering domains. Gaussian Elimination is an example of a linear algebra technique used to solve systems of linear equations efficiently, highlighting the practical applications of these theoretical concepts.

pivot element

The pivot element is a key concept in matrix operations and Gaussian Elimination. A pivot is the leading non-zero element in each row of the matrix after it’s been transformed into RREF. Here's why pivots are important:

• They help in determining the rank of the matrix, which tells us the number of linearly independent rows.
• In solving systems of equations, pivots indicate the variables with clear solutions.
• During matrix simplification, pivots guide the zeroing out of other numbers in their column.

Creating pivots involves adjusting each row systematically until these leading 1s are achieved. Once they're set, they direct further operations that simplify the rest of the matrix, ensuring all other elements in their column are zeros. Mastering the use of pivot elements is essential for students looking to understand matrix reductions.