Question

Use Gaussian Elimination to put the given matrix into reduced row echelon form. $$\left[\begin{array}{cc}4 & 12 \\ -2 & -6\end{array}\right]$$

Short answer

The reduced row echelon form is \( \begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix} \).

Step-by-step solution

Write the augmented matrix

We start by writing the given matrix as an augmented matrix. Here, the matrix is not initially augmented for a system of linear equations, but we handle it similarly:\[\begin{bmatrix} 4 & 12 \ -2 & -6 \end{bmatrix}\]

Make the first pivot equal to 1

To make the pivot of the first row 1, we divide the entire first row by 4:\[\begin{bmatrix} 1 & 3 \ -2 & -6 \end{bmatrix}\]

Eliminate the element below the pivot

We aim to make the element below the pivot in the first column equal to 0 by using row operations. We add 2 times the first row to the second row:\[\begin{bmatrix} 1 & 3 \ 0 & 0 \end{bmatrix}\]

Review for reduced row echelon form

Notice that the second row is all zeros, and the first row already satisfies the criteria for reduced row echelon form (leading 1 in the first column, all elements below leading 1 are 0). Thus, we are done.

Key concepts

Reduced Row Echelon Form

The Reduced Row Echelon Form (RREF) of a matrix is a specific form of a matrix achieved through a series of row operations. In this form, each leading entry in a row (also known as the pivot) is 1, and all other elements in the pivot's column are zeros. Additionally, the pivot of a row is to the right of the pivot in any row above it.
This form is very useful as it simplifies the process of solving systems of linear equations, making it easier to understand the relationship between variables and equations. The process of reaching RREF from a given matrix involves performing several row operations, and the result helps identify unique, infinite, or no solutions to a system of equations.
For instance, in our example, we transformed the given matrix into:\[ \begin{bmatrix} 1 & 3 \ 0 & 0 \end{bmatrix} \]The first row has a leading 1, and the second row, containing all zeros, verifies that the matrix is in RREF.

Matrix Operations

Matrix operations are mathematical procedures used to manipulate matrices to achieve desired results, such as solving systems of equations or finding inverse matrices. The primary matrix operations include addition, subtraction, multiplication, and scalar multiplication.
One key matrix operation from our example is scalar multiplication, where we divided the entire row by a scalar value to achieve a leading 1. This is crucial for simplifying rows and getting them ready for further operations like row addition or subtraction.
Keep these important matrix operation characteristics in mind:

• Commutative for addition: \(A + B = B + A\)
• Not commutative for multiplication: \(AB eq BA\)
• Multiplicative identity: Multiplying by an identity matrix \(I\) leaves the matrix unchanged
• Distributive property: \(A(B + C) = AB + AC\)

Understanding these operations is critical when working to transform matrices as part of Gaussian elimination.

Row Operations

Row operations are the transformations that you can apply to the rows of a matrix to achieve your goals, such as finding the Reduced Row Echelon Form (RREF). They are the building blocks of the Gaussian Elimination method.
Here are the three key types of row operations:

• Row Swapping: Exchanging two rows within the matrix
• Scalar Multiplication: Multiplying a row by a non-zero scalar, useful in setting the pivots to 1. For example, dividing the first row by 4 achieved a leading 1.
• Row Addition: Adding a multiple of one row to another row. This is demonstrated in our example, where 2 times the first row was added to the second row to eliminate the value below the pivot, resulting in:\[ \begin{bmatrix} 1 & 3 \ 0 & 0 \end{bmatrix} \]

Each operation is fundamental in reshaping the matrix without altering the solutions of the corresponding system of equations, explaining their importance in methods like Gaussian Elimination.