QE1 Mars1
the number of variables in a non-homogeneous System AX=Bis, the ranks of A and augmented matrix are p4
and pLA:B respectively, then the system possesses a unique solution if
Answers
Answer:
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Rank and Solutions to Linear Systems
The rank of a matrix A is the number of leading entries in a row reduced form
R for A. This also equals the number of nonrzero rows in R. For any system
with A as a coefficient matrix, rank[A] is the number of leading variables. Now,
two systems of equations are equivalent if they have exactly the same solution
set. When we discussed the row-reduction algorithm, we also mentioned that
row-equivalent augmented matrices correspond to equivalent systems:
Theorem 1.1 If [A|b] and [A0
|b
0
] are augmented matrices for two linear systems
of equations, and if [A|b] and [A0
|b
0
] are row equivalent, then the corresponding
linear systems are equivalent.
By examining the possible row-reduced matrices corresponding to the augmented matrix, one can use Theorem 1.1 to obtain the following result, which we
state without proof.
Theorem 1.2 Consider the system Ax = b, with coefficient matrix A and augmented matrix [A|b]. As above, the sizes of b, A, and [A|b] are m × 1, m × n, and
m × (n + 1), respectively; in addition, the number of unknowns is n. Below, we
summarize the possibilities for solving the system.
i. Ax = b is inconsistent (i.e., no solution exists) if and only if rank[A] <
rank[A|b].
ii. Ax = b has a unique solution if and only if rank[A] = rank[A|b] = n .
iii. Ax = b has infinitely many solutions if and only if rank[A] = rank[A|b] < n.
Step-by-step explanation: