Constructing an Augmented Matrix for a Linear System | Understanding the Relationship between Equations and Variables

if a linear system has n variables and m equations, then the augmented matrix has n rows

In a linear system, we have a set of equations with multiple variables

In a linear system, we have a set of equations with multiple variables. Each equation provides a relationship between these variables. To represent a linear system concisely, we often use an augmented matrix.

An augmented matrix is a matrix that includes both the coefficients of the variables in the equations as well as the constant terms from the right-hand side of the equations. It is used to perform various operations and solve the system efficiently.

To construct the augmented matrix from a linear system, we arrange the coefficients of the variables in each equation as the entries of each row. The constants on the right-hand side of the equations are added as an extra column on the right side of the matrix.

For example, consider a linear system with 3 variables (x, y, and z) and 2 equations:

Equation 1: 2x + 3y – z = 5
Equation 2: x – 2y + 4z = 9

The augmented matrix will have 2 rows (one for each equation) and 4 columns (3 for the variables and 1 for the constant terms):

[ 2 3 -1 | 5 ] –> Equation 1
[ 1 -2 4 | 9 ] –> Equation 2

As you can see from this example, the augmented matrix has n rows, where n is the number of variables in the linear system.

It is important to note that the number of equations in the system (m) may be different from the number of variables (n). In general, there can be three possible outcomes for a linear system:

1. If the number of equations (m) is equal to the number of variables (n), we have a square matrix and the system may have a unique solution, infinitely many solutions, or no solution.

2. If the number of equations (m) is greater than the number of variables (n), the system is overdetermined. In this case, there may be no solution or an approximate solution can be found using methods like least squares.

3. If the number of equations (m) is less than the number of variables (n), the system is underdetermined. In this case, there will be infinitely many solutions.

Therefore, the number of rows in the augmented matrix is determined by the number of equations in the linear system, not the number of variables. The number of columns in the augmented matrix is equal to the number of variables plus 1, accommodating for the additional column for the constant terms.

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