Consistency and solution parameters for systems with given matrix ranks

[lukesta10123]

A set of m linear equations in n unknowns has the m × n matrix A of coefficients
and the m × 1 (column) vector hT of right-hand sides. (Later we shall write this as AxT=hT). T = transpose
In each of cases (a) to (d) below, answer as many as possible of the following questions.
Can the situation occur?
If so, is the set of equations consistent?
If so, how many parameters has the solution?
(a) m = 6, n = 8, r(A) = r(A : hT) = 3 .
(b) m = 7, n = r(A) = r(A : hT) = 3 .
(c) m = 4, n = 5, r(A) = 3, r(A : hT) = 4 .
(d) m = 4, n = 5, r(A) = r(A : hT) = 20 .

Any help on how to approach answering these would be of much help!

very stuck on this.

 

[Mark44]

A set of m linear equations in n unknowns has the m × n matrix A of coefficients
and the m × 1 (column) vector hT of right-hand sides. (Later we shall write this as AxT=hT). T = transpose
In each of cases (a) to (d) below, answer as many as possible of the following questions.
Can the situation occur?
If so, is the set of equations consistent?
If so, how many parameters has the solution?
(a) m = 6, n = 8, r(A) = r(A : hT) = 3 .
(b) m = 7, n = r(A) = r(A : hT) = 3 .
(c) m = 4, n = 5, r(A) = 3, r(A : hT) = 4 .
(d) m = 4, n = 5, r(A) = r(A : hT) = 20 .

Any help on how to approach answering these would be of much help!

very stuck on this.

Your notation might be confusing to some - it would be better to write hT and AxT as h^T and (Ax)^T to better get across the idea that T represents "transpose" and isn't some other matrix.

What does the notation r(A) = r(A : h^T) = 3 mean to you?