Question

Use Gaussian Elimination to put the given matrix into reduced row echelon form. $$\left[\begin{array}{ccc}4 & 5 & -6 \\ -12 & -15 & 18\end{array}\right]$$

Short answer

The matrix in reduced row echelon form is \(\begin{bmatrix} 1 & \frac{5}{4} & -\frac{3}{2} \\ 0 & 0 & 0 \end{bmatrix}.\)

Step-by-step solution

Write Augmented Matrix

We begin by writing the given matrix as an augmented matrix. Since we don't have equalities specified, we consider it as: \[\begin{bmatrix} 4 & 5 & -6 \ -12 & -15 & 18 \end{bmatrix}.\]

Make Leading Coefficient of Row 1 a 1

To make the leading coefficient of Row 1 a 1, divide the entire first row by 4:\[\begin{bmatrix} 1 & \frac{5}{4} & -\frac{3}{2} \ -12 & -15 & 18 \end{bmatrix}.\]

Eliminate First Element of Row 2

To eliminate the first element of the second row, add 12 times the first row to the second row:Original Row 2: \(-12, -15, 18\)New Row 2: \((-12) + 12(1), (-15) + 12(\frac{5}{4}), 18 + 12(-\frac{3}{2})\) which results in:\[\begin{bmatrix} 1 & \frac{5}{4} & -\frac{3}{2} \ 0 & 0 & 0 \end{bmatrix}.\]

Verify Row Echelon Form

The matrix \[\begin{bmatrix} 1 & \frac{5}{4} & -\frac{3}{2} \ 0 & 0 & 0 \end{bmatrix}\] is now in reduced row echelon form (RREF) as all criteria are met: leading coefficients are 1 and each leading one is to the right of the leading one in the row above. All rows of zeros are at the bottom.

Key concepts

Reduced Row Echelon Form

Reduced Row Echelon Form (RREF) is a form of a matrix that makes solving systems of linear equations very easy. It is a specific form where each leading entry (or pivot) in the rows is 1, and all the other entries in that column are zeros. One row operation that can help achieve this is called Gaussian elimination. This involves:

• Swapping rows.
• Multiplying rows by non-zero constants.
• Adding or subtracting multiples of rows from each other.

Once in RREF, the matrix simplifies determining solutions to the system of equations or understanding the properties of the linear system. For example, notice how in the final step of the provided solution, the matrix reached RREF once the leading numbers (also known as pivots) were 1, and the rows below were zeros.

Matrix Algebra

Matrix algebra involves operations with matrices—arrays of numbers arranged in rows and columns. These operations include addition, subtraction, multiplication, and more complex tasks like finding inverses or determinants. Matrix algebra helps in transforming matrices into simpler forms for analysis or solving equations.
For example, in the given solution, the process of Gaussian elimination involves matrix algebra techniques like scaling a row by a constant and adding one row to another. By understanding and applying these operations properly, matrices can be simplified to forms like RREF to make interpretation easier. Learning matrix algebra is fundamental for fields ranging from engineering to computer science where systems of equations frequently arise.

Augmented Matrix

An augmented matrix combines the coefficients of a system of linear equations and their constants into a single matrix. This is particularly useful when applying methods like Gaussian elimination to find solutions. It provides a concise way of displaying information and facilitating row operations without repeatedly rewriting the entire set of equations.
In the exercise solution, an augmented matrix was formed by arranging coefficients of the linear equations. Although not initially given in equation form, the matrix was considered as already augmented based on its elements. This sets the stage for carrying out elimination procedures to rearrange it into reduced row echelon form.