Reduced Row Echelon Form My personal favorite Elementary
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Reduced Row Echelon Form My personal favorite!
Elementary Row Operations We can perform three elementary row operations on matrices: • Multiplying a row by a constant. • Switching two rows. • Adding a constant times a row to another row. To solve a system of equations, first put into augmented matrix form. This system 4 x – 5 y + 3 z = 2 x – y – 2 y = -6 4 x – 4 y – 14 z = 18 BECOMES
To row reduce a matrix: 1. Perform elementary row operations to yield a "1" in the first row, first column. 2. Create zeros in all the rows of the first column except the first row by adding the first row times a constant to each other row. 3. Perform elementary row operations to yield a "1" in the second row, second column. 4. Create zeros in all the rows of the second column except the second row by adding the second row times a constant to each other row. 5. Perform elementary row operations to yield a "1" in the third row, third column. 6. Create zeros in all the rows of the third column except the third row by adding the third row times a constant to each other row. Continue this process until the first m×m entries form the identity matrix.
TA DAH! Notice the identity matrix in the first 3 rows and columns. Now we are ready to read the results! Putting this matrix back into equation form: 1 x + 0 y + 0 z = -123, x = -123 0 x + 1 y + 0 z = -103 y = -103 0 x + 0 y + 1 z = -7 (-123, -103, -7) is the ordered triple solution.
Reduced Row Echelon Form (RREF) is on your calculator too. Let’s try another problem, this time doing things by calculator: x – 2 y + 3 z = 9 -x + 3 y = -4 2 x – 5 y + 5 z = 17 Put into calculator as 3 x 4 matrix. Go to main screen, go to matrix menu, choose math, RREF (matrix name) I love it!
This is method gives you the most info, even if the system comes out with no solution or an infinite number of solutions:
This method is so wonderful that someone has already built an App that makes use of it. If you have PLY SMLT 2 you are ready to do a bunch easily. If not, you will have to create an appropriately sized augmented matrix, perform RREF on the matrix, and life is still good!
