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# Matrices and equations

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1. Guassian Elimination is really just using the regular Elimination Method and applying the Triangular Form procedures to a set of linear equations, but without having to write down all the variables in the equations each time. This in itself simplifies the procedure and helps focus on the task. What is the ultimate goal of all these methods and procedures?

2. If you had 5 linear equations using 5 variables, what would the augmented matrix look like? What would this matrix look like in echelon form?

3. What does the final row of the augmented matrix in echelon form (the above description) tell you about the solution to the system of 5 linear equations?

4. Is it always possible to take any square matrix and write it in echelon form with all 1's along its main diagonal?

5. Can you take any two matrices and add them, subtract them and multiply them together?

6. Is it possible to divide two matrices?

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1. Guassian Elimination is really just using the regular Elimination Method and applying the Triangular Form procedures to a set of linear equations, but without having to write down all the variables in the equations each time. This in itself simplifies the procedure and helps focus on the task. What is the ultimate goal of all these methods and procedures? Guassian elimination will allow us eliminate unknowns, which will in turn led to us arriving at a system that is easily solvable. This procedure is used to determine whether or not the set of linear equations will have unique solution, no solution, infinitely many solutions. [Unique solution is defined as there exist one and only one set of values that satisfied the set of linear equations simultaneously. No solution is defined as no set of values for the variables in the set of linear equations that can satisfies all equations i.e. 0xi = a real number (1, infinity). Infinitely many solutions are there are many sets of values that can satisfy your set of linear system!]. Gaussian elimination can be used to describe the set of all solutions if there exist many solutions that can satisfy the equations.

2. If you had 5 linear equations using 5 variables, what would the augmented matrix look like? What would this matrix look like in echelon form? Define A to be a matrix of 5x5 and B is the solution of those 5 linear equations. Our augmented matrix will be [A|B]⇒ . Supposed that your linear equations will be have unique solution, that is every row that is not [a11,...,a55] will be zero, your Echelon form of this matrix will be , where * may have any values.

3. What does the final row of the augmented matrix in echelon form (the above description) tell you about the solution to the system of 5 linear equations? B is usually a matrix consists of the constraints of any system of linear equations. In other words, it is the solution of each linear system. Supposed that B is a 5x1 matrix, after row-reducing(aka. Gaussian method), you obtain 1's in [a11,...,a55], then the last column in the row reducing equations will be the solution of variables in the set of linear equations.

4. Is it always possible to take any square matrix and write it in echelon form with all 1's along its main diagonal? It is possible. This is done after you perform Gauss-Jordan elimination method. In other words, your last row will contains 0,....,1. Then you perform reverse addition to make sure that all variable except the pivots to be zero!

5. Can you take any two matrices and add them, subtract them and multiply them together? You can add and subtract any two matrices together as long as they have the same mxn dimension. You can multiply any two matrices together as long as the matrices conform with each other.

6. Is it possible to divide two matrices? Division is not defined for any two matrices! However, you can invert matrices. Although this is similar, it is NOT the same as dividing!

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