Solving systems using matrices is a powerful technique in linear algebra. It allows us to represent multiple equations compactly and solve them efficiently. By transforming augmented matrices through row operations, we can find solutions to complex systems of equations.

This method connects to the broader study of matrices by showcasing their practical applications. It demonstrates how matrix operations can be used to solve real-world problems, from economics to engineering, by representing and manipulating systems of linear equations.

Augmented matrices for systems

Representing systems of linear equations

• A system of linear equations consists of two or more linear equations with the same variables
• An augmented matrix represents a system of linear equations
  • Formed by writing the coefficients of the variables as the entries in a matrix
  • Constants on the right side of the equations are written as the last column
• The number of rows in the augmented matrix equals the number of equations in the system
• The number of columns in the augmented matrix equals the number of variables plus one (for the constants)
• When writing the augmented matrix, ensure the variables are written in the same order for each equation and the coefficients are in the corresponding matrix positions

Dimensions and structure of augmented matrices

• The dimensions of the augmented matrix depend on the number of equations and variables in the system
  • For a system with $m$ equations and $n$ variables, the augmented matrix will have $m$ rows and $n+1$ columns
  • Example: A system with 3 equations and 2 variables will have a $3 \times 3$ augmented matrix
• The structure of the augmented matrix follows a specific pattern
  • The first $n$ columns contain the coefficients of the variables
  • The last column contains the constants from the right side of the equations
  • Example: For the system $2x + 3y = 5$ and $4x - y = 3$, the augmented matrix would be $\begin{bmatrix} 2 & 3 & 5 \ 4 & -1 & 3 \end{bmatrix}$

Solving systems with row operations

Elementary row operations

• Elementary row operations transform the augmented matrix into row-echelon or reduced row-echelon form
  • Row-echelon form: Matrix where all rows with only zeros are at the bottom and the leading entry (first non-zero number from the left) of a row is always strictly to the right of the leading entry of the row above it
  • Reduced row-echelon form: Row-echelon form where leading entries are 1 and the column containing the leading 1 has zeros in all its other entries
• There are three types of elementary row operations:
  • Type 1: Interchange two rows
  • Type 2: Multiply a row by a non-zero constant
  • Type 3: Add a multiple of one row to another row
• Goal is to transform the matrix into a form where the solution can be easily read, typically with ones on the diagonal and zeros above and below the diagonal

Applying row operations strategically

• The order in which elementary row operations are performed matters
  • Operations should be chosen strategically to efficiently transform the matrix into the desired form
  • Example: First, use Type 3 operations to eliminate coefficients below the diagonal, then use Type 2 operations to make leading entries equal to 1
• The solution to the system of equations can be read from the final row-echelon or reduced row-echelon form of the matrix
  • The values of the variables correspond to the positions of the leading coefficients in the matrix
  • Example: In a reduced row-echelon form $\begin{bmatrix} 1 & 0 & 3 \ 0 & 1 & -2 \end{bmatrix}$, the solution is $x = 3$ and $y = -2$

Consistency and independence of systems

Determining consistency

• A system of linear equations can be classified as consistent or inconsistent based on whether it has a solution
  • Consistent system: Has at least one solution
  • Inconsistent system: Has no solution
• In the row-echelon or reduced row-echelon form of the augmented matrix, a consistent system will have no rows where all entries are zero except for the last entry
  • A row with all zeros except for the last entry would imply an equation of the form $0 = c$, where $c$ is a non-zero constant, which has no solution
  • Example: A row-echelon form $\begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & 4 \ 0 & 0 & 1 \end{bmatrix}$ represents a consistent system, while $\begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & 4 \ 0 & 0 & 0 \end{bmatrix}$ represents an inconsistent system

Determining independence

• A system of linear equations can be classified as independent or dependent based on the number of solutions it has
  • Independent system: Has a unique solution
  • Dependent system: Has infinitely many solutions
• In the reduced row-echelon form of the augmented matrix, an independent system will have a leading coefficient (a one) in every column except possibly the last
  • A dependent system will have at least one column without a leading coefficient
  • Example: A reduced row-echelon form $\begin{bmatrix} 1 & 0 & 3 \ 0 & 1 & 4 \end{bmatrix}$ represents an independent system, while $\begin{bmatrix} 1 & 0 & 3 \ 0 & 0 & 0 \end{bmatrix}$ represents a dependent system

Interpreting solutions in context

Modeling real-world situations

• Systems of linear equations can model a wide variety of real-world situations (business problems, scientific experiments, engineering designs)
• Variables in the system represent unknown quantities in the real-world situation
• Equations represent relationships between these quantities
• The solution to the system represents the values of the unknown quantities that satisfy all relationships simultaneously
  • Example: In a business problem, $x$ could represent the number of units produced, $y$ the number of units sold, and the equations could represent the relationship between production cost, revenue, and profit

Interpreting solutions and their implications

• The solution to the system may not always make sense in the context of the real-world situation
  • Example: If the solution involves negative quantities when only positive quantities are possible, the model may need to be modified or the solution interpreted differently
• The number of solutions to the system can also have implications for the real-world situation being modeled
  • No solution may indicate that the relationships being modeled are inconsistent or impossible to satisfy simultaneously
  • Infinitely many solutions may suggest that there is insufficient information to determine a unique solution or that some quantities can vary independently
  • Example: In a system modeling the number of products to manufacture, no solution could imply that the production constraints are impossible to meet, while infinitely many solutions could suggest that there are multiple production plans that satisfy the constraints