Call Now: (800) 537-1660
 Home    Why?     Free Solver   Testimonials    FAQs    Product    Contact

May 22nd

May 22nd

# Solving Linear Equations

One method to put a matrix A into reduced row echelon form (RRE):

NOTE: The leading entry of a row of a matrix is the left-most non- zero entry of that row.

1. Be clever. Then, put all rows consisting entirely of zeros at the bottom of the matrix.
2. Choose a simple row with a leading entry in the first column. Switch this row with the first
row. Use this row and elementary row ope rations to get all entries in the first column below
the first row equal to zero. (I say clean down the first column).
column. Switch this row with the second row. Use this row and elementary row operations to
make
all entries be low its leading entry equal to zero (clean down the second column).
4. Keep cleaning down successive columns.
The matrix is now in row echelon form (but not necessary reduced row echelon form). The
leading entries are in the pivot positions for A.
5. Clean up. That is, use the lowest non-zero row and elementary row operators to make all
entries in the column above its leading entry zero. Make that leading entry equal to one using
a row
operation.
6. Now, forget about this lowest row and use the row above it to clean up, making all entries in
the column above its leading entry zero. Then, make its leading entry equal to one using a
row operation.
7. Continue cleaning up. Then, make sure all leading entries are equal to one and the matrix is
in RRE.

To solve m × n linear algebraic equations Ax = b:

1. Form the augmented matrix [Ab] and reduce to RRE.
2a. If in this process you get a leading entry in the last column (the matrix has a row: (0, . . . , 0, b)
where b ≠ 0) then the system is inconsistent . (That is, there are no solutions to the equations
because one of the equations is 0 = b.)
2b. If not, continue to reduce to RRE, convert to equations and solve for the leading variables (or
basic variables) ( variables corresponding to leading entries of the reduced matrix) in terms of
the free
correspond to pivot columns of A.

To solve the m × n homogeneous linear algebraic equations Ax = 0:

1. Reduce the augmented matrix [A0] to RRE. (You can reduce A as long as you convert back
to homogeneous equations.)
2. Solve for the basic (leading) variables in terms of free variables (if there are free variables).
3. If there are free variables: successively set each free variable equal to one (or an ‘easy’ non-
zero number ) and set the other free variables equal to zero. Then, write your solutions as
vectors. This gives a list of linearly independent solution vectors u1, . . . ,um(one for each
free variable). Every solution to Ax = 0 is a linear combination of these solution vectors;
therefore, the solution set to Ax = 0 is span{u1, . . . ,um}.

 Prev Next

Home    Why Algebra Buster?    Guarantee    Testimonials    Ordering    FAQ    About Us
What's new?    Resources    Animated demo    Algebra lessons    Bibliography of     textbooks