Ace college algebra II notes
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- Navigate this page:
- Augmented Matrices
- Solving a System Using a Matrix
- Matrix Row Operations
- Gaussian Elimination
- Gaussian Elimination with Back-Substitution
- Gauss-Jordan Elimination
- Using TI-Nspire to Find Reduced Row-Echelon Form
If you’ve ever worked with a program like Microsoft Excel®, then you most likely already know what a matrix is! It is simply an array of values organized into rows and columns. 
Augmented Matrices A matrix can give us a shortened way of writing a system of equations. The first step in solving a system of equations using a matrix is to write an augmented matrix. Here is an example:
Write the solution set for a system of equations represented by the matrix .
The following row operations produce matrices that represent systems with the same solution set: 1. Two rows of a matrix may be interchanged. This is the same as interchanging two equations in a linear system. 2. The elements in any row may be multiplied by a nonzero number. This is the same as multiplying both sides of an equation by a nonzero number. 3. The elements in any row may be multiplied by a nonzero number, and these products may be added to the corresponding elements in any other row. This is the same as multiplying both sides of an equation by a nonzero number and then adding equations to eliminate a variable. Two matrices are row equivalent if one can be obtained from the other by a sequence of row operations. Performing Matrix Row Operations Example 2 Use the matrix and perform each indicated row operation: 
a.  b.  c. 
Gaussian Elimination The process that we use to solve linear systems using matrix row operations is called Gaussian elimination after German mathematician Carl Friedrich Gauss (1777-1855). Solving Linear Systems Using Gaussian Elimination 1. Write the augmented matrix for the system. 2. Use matrix row operations to simplify the matrix to a row-equivalent matrix in row-echelon form, with 1’s down the main diagonal from upper left to lower right, and 0’s below the 1’s. 
3. Write the system of linear equations corresponding to the matrix in Step 2 and use back substitution to find the system’s solution. Get 1 in the upper left-hand corner. Use the 1 in the first column to get 0’s below it. Get 1 in th second row, second column positio0n. Use the 1 in the second column to get 0 below it. Get 1 in the third row, third column position. Gaussian Elimination with Back-Substitution Example 3 Use matrices to solve the system: 
Gauss-Jordan Elimination When we use Gaussian elimination, we obtain a matrix is row-echelon form, with 1’s down the main diagonal and 0’s below the 1’s. A second method, called Gauss-Jordan elimination, after Carl Friedrich Gauss and Wilhelm Jordan (1842-1899), continues the process until a matrix with 1’s down the main diagonal and 0’s in every other position above and below the 1’s is found. Such a matrix is said to be in reduced row-echelon form and looks like this: 
Modern supercomputers are capable of solving systems with more than 600,000 variables. The augmented matrices for such systems are huge, but the process the computers follow is very much like we are doing by pencil and paper. Using TI-Nspire to Find Reduced Row-Echelon Form Begin with the Home Screen Press 7: Matrix & Vector and choose Caclulate 1: Create --------------------------> 1:Matrix   
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