Matrices and Linear Systems
Math 260, Applied Linear Algebra and Matrices, Fall 2005
J. Gerlach
This is an overview regarding the results for
the reduced row echelon form, for singular or
non-singular matrices and their relations to linear systems. In the statements below A and C
are matrices, and B, D and X are vectors. We
assume that the matrices and vectors have the
appropriate sizes.
Elementary Row Operations:
are called a linear system. If B = 0 we speak of
a homogeneous system. A homogeneous system
always has the trivial solution X = 0.
Theorem 1.7: If the augmented matrices
[A B] and [C D] are row equivalent, then the
systems AX = B and CX = D have exactly
the same solution set.
Corollary 1.1: If A and C are row equivalent, then AX = 0 and CX = 0 have exactly
the same solutions.
Definition: A square matrix A is nonsingular, if it has an inverse matrix A−1 , such
that
AA−1 = A−1 A = I
1. Interchange rows.
2. Multiply a row by a non-zero constant.
3. Add a multiple of one row to another row.
Row Equivalent Matrices: Two matrices
A and B are row equivalent if they can be obtained from each other by a finite sequence of
elementary row operations. (p. 46)
Reduced Row Echelon Form (rref ): A
matrix is in reduced row echelon form if
Theorem (1.12, 1.13, 1.14): Let A be a
square matrix. Then the following are equivalent:
1. A is non-singular.
1. Rows of zeros are at the bottom.
2. A is row equivalent to the identity matrix,
i.e. rref (A) = I.
2. The first non-zero entry in each row is one
(called leading one).
3. X = 0 is the only solution to AX = 0.
3. The leading one in a given row is to the
right of the leading one of the row directly
above it.
4. AX = B has a exactly one solution X for
every B.
4. Columns containing a leading one have zeros in the other entries.
Corollary: If A is a singular square matrix,
then AX = B either has no solution, or it has
infinitely many solutions.
Theorem 1.6: Every matrix A is row equivalent to exactly one matrix C in reduced row
echelon form. Notation:
Theorem (T.1.6.13): Let Xp be a particular solution to AX = B, i.e. AXp = B, and let
Xg be an arbitrary solution of AX = B. Then
Xg has the form Xg = Xp + Xh , where Xh is a
solution of the homogeneous system AX = 0.
C = rref (A)
Let A be a matrix, and B and X be vectors,
then the equations
AX = B
1