91Ó°ÊÓ

Gaussian elimination is a method used to solve systems of linear equations. It involves using row operations to simplify the augmented matrix of the system into an upper triangular form or even further into a simpler form called row echelon form (REF) or reduced row echelon form (RREF). These row operations include:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting a multiple of one row to another row

In the context of our exercise, Gaussian elimination helps us transform the augmented matrix to a simpler form to identify values of the parameter that make the system dependent. This technique is foundational in linear algebra because it offers a structured way to handle multiple equations with multiple variables.

An augmented matrix is a compact way of representing a system of linear equations. Each row in the augmented matrix corresponds to an equation, and each column corresponds to a coefficient of a variable or the constants on the right-hand side of the equations.

Take for example, the system:

\begin{aligned} 5x - 6y + kz &= -5 \ x + 3y - 2z &= 2 \ 2x - y + 4z &= -1 \ \text{can be written as} \ \text{augmented matrix} \ \begin{array}{ccc|c} 5 & -6 & k & -5 \ 1 & 3 & -2 & 2 \ 2 & -1 & 4 & -1 \ \right.\right.\right. \text{This representation} \ allows us to easily apply row operations \ \text{which simplifies the system.} \text{} \text{}Returning} \text{} \text{}\right./i\right. to our exercise; the aim} \ \text{is to transform} the augmented matrix \ \begin{array}{ccc|c} 5 & -6 & k & -5 \ 1 & 3 & -2 & 2 \ 2 & -1 & 4 & -1 \ \right.\right.\right. to a simpler form \ \ text{} and solve for} the variable \ \(k\).

The reduced row echelon form (RREF) of a matrix is a unique form of a matrix achieved by using row operations where each leading entry in a row is 1 and is the only non-zero entry in its column. Moreover, each leading 1 is the only non-zero entry in its column above it and below it.

Here are the properties of RREF:

• Any row with all zero entries is at the bottom of the matrix
• The first non-zero entry in any row is 1 (leading 1)
• Each leading 1 is to the right of the leading 1 in the row above it
• The leading 1 is the only non-zero entry in its column

In our exercise, we use Gaussian elimination to transform the original augmented matrix into RREF. This process reveals the relationship between the parameter \(k\) and the other coefficients, telling us for which \(k\) the system becomes dependent. In our case, solving for \(k\) in the context of the reduced form led us to \(k = -4\). This value ensures one of the rows becomes a linear combination of the others, indicating a dependent system.