Download Solution for Linear Systems

Step 2: Convert
into row reduced echelon form
Step 3: If
, then the system is consistent. Otherwise inconsistent.
Step 4: If
is consistent, then solution is obtained from the echelon form of
Note: If
, then there will be
and remaining variables are dependent on
.
variables which are Linearly Independent
variables
Homogeneous system of Equations
The system of equations
i.e.
is said to be homogeneous system of equations if
.
To obtain solution of homogeneous system of equations the procedure is as follows:
Step 1: Convert
into row reduced echelon form
Step 2: Depending on nature of
, we will solve further.

is always consistent.

has a Trivial solution always (i.e. Zero solution)
 If
, (number of variables), then
 If
then
 If
has Unique solution.(Trivial solution)
has only Trivial solution i.e. Zero Solution
(number of variables (or) unknowns), then
has infinitely many
solutions.
 If
, then
has Infinitely many solutions.
Matrix Inversion Method
Suppose
is a non-homogeneous System of equations, such that
and
, then
and
, then
has unique solution and is given by
Cramer’s Rule
Suppose
the solution of
is a non-homogeneous System of equations, such that
is obtained as follows:
Step 1: Find determinant of
i.e.
(say)
Step 2: Now,
where
is the determinant of
by replacing 1st column of
with .
Step 3: Now,
where
is the determinant of
by replacing 2nd column of
with .