12.5 Row Canonical Form

To reduce a matrix to row canonical form, (AKA row reduced echelon form, or “reduced row-echelon” form, or Gauss-Jordan form), first reduce it to echelon form using Gaussian elimination as described in section 12. Then, starting from the last equation with a nonzero coefficient of an unknown,

1.

Use the equation to eliminate the first unknown with a nonzero coefficient in the equation, (i.e. the pivot unknown), from the previous equations (i.e. create zeros above the pivot). Do not worry about fractions anymore; they cannot normally be avoided.

2.

Divide the equation by the pivot so that the pivot becomes 1.

3.

Go to the previous equation, if any, and repeat the process.

Note: you can do both steps 1 and 2 above in each stage at the same time. You do not have to show each row nor the final scaling using a separate matrix. You must however explicitly show the multiplier(s) used.

Note: in this class you may not reduce a matrix to row canonical unless you are explicitly told so or the given class procedure requires it. Normally you must stop after you did the Gaussian elimination (producing normal echelon form with pivots that in general are not 1). That is because this is the standard and most efficient way to do it.

You may however proceed to row-canonical when finding the inverse of a matrix using the Gaussian Elimination method described later. You must proceed to row-canonical to find a simplified basis of the row space of a matrix, or of the column space, (starting from the transpose matrix), as described later.