Matricies

Augmented Matrix: A matrix where each row is an equation, and each column is the coefficients for a particular variable in each equation. For example:

$$\begin{array}{rl}6x_1+x_2-3x_4& =11\\ 3x_2-4x_3+5x_4& =6\\ 2x_3-4x_4& =-3\end{array}\to \left[\begin{array}{cccccc}6& 1& 0& -3& |& 11\\ 0& 3& -4& 5& |& 6\\ 0& 0& 2& -4& |& -3\end{array}\right]$$

Gaussian Elimination: Using elementary row operations to put a matrix in echelon form

Gauss-Jordan Elimination: Using elementary row operations to put a matrix in reduced echelon form

Reduced Echelon Matrix: A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions:

1. It is in row echelon form.
2. The leading entry in each nonzero row is 1 (called a leading one).
3. Each column containing a leading 1 has zeros in all its other entries.
   Steps:
4. Make $a_{11}$ 1
5. Make everything below $a_{11}$ zero
6. Repeat with $a_{22}$

Free Variable: A variable that can’t be eliminated or constrained. Happens when you have more variables than equations.

Elementary Row Operations

Equivalent to operations on systems
Interchange Equations: Swap two rows

Multiply an Equation by a Constant: Multiply one row by a constant.. Ex:

$$\left[\begin{array}{cccccc}6& 1& 0& -3& |& 11\\ 0& 3& -4& 5& |& 6\\ 0& 0& 2& -4& |& -3\end{array}\right]\to \left[\begin{array}{cccccc}0& 3& -4& 5& |& 6\\ 6& 1& 0& -3& |& 11\\ 0& 0& 2& -4& |& -3\end{array}\right]$$

Add a multiple of one equation to another:
Ex: Row 2 + 2* Row 3 $\to$ Row 2

$$\left[\begin{array}{cccccc}6& 1& 0& -3& |& 11\\ 0& 3& -4& 5& |& 6\\ 0& 0& 2& -4& |& -3\end{array}\right]\to \left[\begin{array}{cccccc}6& 1& 0& -3& |& 11\\ 0& 3& 0& -3& |& 0\\ 0& 0& 2& -4& |& -3\end{array}\right]$$

Application/Problem Types

Wire Temperature

1. Make an equation for each node, temp = average of all connected nodes
2. solve for constant
3. Put into matrix
4. RREF

Balance the Chemical Equation

Traffic Management:

1. Write an equation for each intersection (A,B,C) where one side is the sum of all the input and the other is the side of all the output
   A: $40+30+x_4=50+x_1$
2. Solve equations for the constant
3. RREF

Find the missing values in the partial fraction decomposition

1. Generate a single, distributed equation
2. Separate each “set of coefficients” (here it’s constants and coefficients of x, but it could be powers of x or different variables (I think))
3. RREF