# Homework Help: Question about the properties of the rank of a matrix

1. Mar 13, 2012

### skyturnred

1. The problem statement, all variables and given/known data

Matrix A is a 4 row by 5 column matrix. Matrix B is a column vector in R$^{4}$. We are supposed to decide whether the following are (a) no solution, (b) one solution, or (c) infinitely many solutions, or whether (d) the data do not give enough information to tell, or (e) the data are impossible.

For the following six cases:

rank(A) rank([A | b])
(i) 4 5
(ii) 4 4
(iii) 4 3 .
(iv)3 4
(v) 3 5
(vi)3 3

***numbers on the right are the rank for augmented matrix, numbers on left are for matrix A alone

2. Relevant equations

3. The attempt at a solution

I THINK I know the answer to all of them.. but I am not entirely certain. I have always found rank just a little confusing. Can someone please tell me where I went wrong in the following and why?

(i) The data is impossible: rank exceeds number of rows.
(ii) There are 5 unkowns (because matrix A has 5 columns) and the rank is 4. So that means there is ONE free variable, so there are infinitely many solutions.
(iii) Not possible: augmenting Matrix A by a column vector B cannot possible DECREASE the rank. Can only be equal to or ONE more than rank(A).
(iv) No solutions. Since augmenting by a column vector increased the rank by one, that means that the augmented matrix is inconsistent.
(v)Not possible: rank exceeds number of rows
(vi)Infinitely many solutions: 5 unknowns, rank is 3, so 2 free variables.

Thanks!