Problem 52 - College Algebra

Question

Describe what is meant by the augmented matrix of a system of linear equations.

Step-by-Step Solution

Answer

An augmented matrix is a matrix formed by appending the columns of constants on the right side of the equal symbol, to the coefficient matrix of a system of linear equations. The two are separated by a vertical bar.

Step 1: Describing Basic Concept of Matrices

A matrix is essentially an array of numbers written within square brackets. When dealing with systems of linear equations, each row in the matrix represents an equation, and each column represents a coefficient of a variable.

Step 2: Describing the Coefficient Matrix

In a system of linear equations, the coefficient matrix specifically contains the coefficients of the variables in the system. For instance, in the equations \(2x + 3y = 6\) and \(4x - y = 8\), the coefficient matrix would be \(\begin{bmatrix} 2 & 3 \ 4 & -1 \end{bmatrix}\) .

Step 3: Describing the column of constants on the right

We also have the constants on the right side of the equations, for the same system of equations, the right-side constants are [6,8].

Step 4: Describing Augmented Matrix

The augmented matrix combines the coefficient matrix and the column of constants into a single matrix. By using a vertical bar to separate the coefficients from the constants, we arrive at the augmented matrix of the system. For the previous system of equations it will be: \(\begin{bmatrix} 2 & 3 | & 6 \ 4 & -1 | & 8 \end{bmatrix}\) .