In: Math
Suppose C is a m × n matrix and A is a n × m matrix. Assume CA = Im (Im is the m × m identity matrix). Consider the n × m system Ax = b.
1. Show that if this system is consistent then the solution is unique.
2. If C = [0 ?5 1
3 0 ?1]
and A = [2 ?3
1 ?2
6 10] ,,
find x (if it exists) when
(a) b =[1
0
3]
(b) b =[ 7
4
22] .
Given, C is a m×n matrix and A is a n×m matrix and CA = Im, where Im is the m×m identity matrix.
1) Given system is Ax = b.
Multiplying C on both sides we get, C(Ax) = Cb
i.e., (CA)x = Cb
i.e., x = Cb
Since Cb is an unique mx1 matrix, therefore Ax = b has an unique solution.
Hence, the given consistent system has an unique solution.
2) Given, C = and A =
Now, CA = =
(a) Here b = .
Now, x = Cb
i.e., x =
i.e., x = .
(b) Here x =
Now, x = Cb
i.e., x =
i.e., x = .