This question is about reduced row echelon form, Gauss-Jordan, inverting matrices, and solving systems of equations.
I try to solve a system of equations with matrices. I know what operations are allowed, but I just seem to arrive at the wrong conclusion 50 % of the times. So here are three problems, each with my calculation. My hope is to clarify if I:
- am making a careless misstake, and where those mistakes are (if so, I may have to do these problems in a slower pace)
- do not know the theory well enough (don't make the correct steps)
- use a bad or "not smart" way of attacking the problem. (for example, if I do row1 + row2 when I shoul have taken row1 - row3).
problem 1
problem: see picture.
solution: see picture.
problem 2
problem: see picture.
solution: see picture.
problem 3
problem:
solution: see picture below.
I say $x_3=-\frac{1}{a^2-2}$ while the book says $x_3=-a^2-2$
You have to understand the rules of row echelon form and reduced row echelon form before tackling these problems. First we need a leading one on the top left corner. Then the leading one must go down and shift to the right If applicable, a row of zeros on the bottom of the matrix.
For reduce row echelon form all of these rules apply along with this one: the leading one must have 0's on the top and bottom of the column.
Now there are three row operations: switching the rows multiplying a row with a number adding two rows and use the new result to replace the old row The main diagonal is on a11, a22, and a33 . I like to draw triangles as a border line. For row echelon form, you need and upper triangular matrix. Therefore, the numbers that are below the main diagonal need to be turned to zero.