State whether each of the following sets of data are possible for the matrix equation ax = b. if possible, describe the solution set. that is, tell whether there exists a unique solution, no solution or infinitely many solutions. here, [a|b] denotes the augmented matrix.

(a) a is a 5 × 6 matrix, rank(a) = 4 and rank[aib] = 4.
(b) a is a 3 x 4 matrix, rank (a) = 3 and rank [aib] = 2.
(c) a is a 4 × 2 matrix, rank (a) = 4 and rank[a|b| = 4
(d) a is a 5 × 5 matrix, rank (a) = 4 and rank [a|b] = 5.
(e) a is a 4 × 2 matrix, rank (a) = 2 and rank [a|b] = 2.

(A) Possible, the system will always have infinite solutions.

(B) Impossible.

(C) Impossible.

(D) Possible, there is no solutions.

(E) Possible, unique solution.

Step-by-step explanation:

(A) It possible, for example consider

in this case, the system will always have infinite solutions (there at most two variable that could take any valour).

(B) It is impossible because .

(C) It is impossible because .

(D) It is possible, for example consider

but in this case there is no solutions.

(E) It is possible, for instance, consider

Now, if this is the case there exists a unique solution.