(1) If the first column is all zeros, “mentally cross it off.” Repeat this
process as necessary.
(2a) Use row interchange if necessary to get a nonzero entry (pivot) p in
the top row of the remaining matrix.
(2b) For each row R below the row containing this entry p, add −r/p
times the row containing p to R where r is the entry of row R in the
column which contains pivot p. (This gives all zero entries below pivot p.)
(3) “Mentally cross off” the first row and first column to create a smaller
matrix. Repeat the process (1) - (3) until either no rows or no columns
Example. Page 69 number 16. (Put the associated augmented matrix
in row-echelon form and then use substitution.)
Note. The above method is called Gauss reduction
with back substitution.
Note. The system
is equivalent to the
where is the ith
column matrix of A. Therefore,
if and only if is in the span of
(the columns of A).
Definition. A matrix is in reduced row-echelon form
if all the pivots
are 1 and all entries above or below pivots are 0.
Example. Page 69 number 16 (again).
Note. The above method is the Gauss-Jordan method.
Theorem 1.7. Solutions of
be a linear system and let where H is in
(1) The system
is inconsistent if and only if
has a row
with all entries equal to 0 to the left of the partition and a nonzero entry
to the right of the partition.
(2) If is consistent
and every column of H contains a pivot, the
system has a unique solution.
(3) If is consistent
and some column of H has no pivot, the
system has infinitely many solutions, with as many free variables as there
are pivot-free columns of H.
Definition 1.14. A matrix that can be obtained from an identity matrix
by means of one elementary row operation is an elementary matrix.
Theorem 1.8. Let A be an m × n matrix and let E be an m × m
elementary matrix. Multiplication of A on the left by E effects the
same elementary row operation on A that was performed on the identity
matrix to obtain E.
Proof for Row-Interchange. (This is page 71 number 52.) Suppose
E results from interchanging rows i and j:
Then the kth row of E is [0, 0, . . . , 0, 1, 0, . . . ,
(1) for the nonzero
entry if the kth entry,
(2) for k = i the nonzero entry is the jth entry, and
(3) for k = j the nonzero entry is the ith entry.
Let , and .
The kth row of B is
Now if then all
are 0 except for p = k and
Therefore for , the kth
row of B is the same as the kth row of
A. If k = i then all are 0 except for p = j
and the ith row of B is the same as the jth row of A.
Similarly, if k = j
then all are 0 except for p = i and
and the jth row of B is the same as the ith row of A.
Example. Multiply some 3 × 3 matrix A by
to swap Row 1 and Row 2.
Note. If A is row equivalent to B, then we can find C such that CA = B
and C is a product of elementary matrices.
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