Consider the real system of matrix A, determine all solutions.

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



Consider the real system:



[tex]

\begin{pmatrix}

1 & 2 & 3 & 4\\

4 & 3 & 2 & 1\\

-5 & -5 & -5 & -5

\end{pmatrix}x=\begin{pmatrix}a\\b\\c\end{pmatrix}

[/tex]



and denote the system by matrix A.



1. What is the rank of A? By inspection, determine a non-zero vector in the null space of A-transpose.

2. Under what conditions on the numbers a, b, c is the system solvable? (Use your solution of part a).

3. Determine all solutions of the system for a = b = 1, c = -2.



2. Relevant equations



No relevant equations



3. The attempt at a solution



For part 1, I reduced to row echelon and got:



[tex]

\begin{pmatrix}

1 & 0 & -1 & -2\\

0 & 1 & 2 & 3\\

0 & 0 & 0 & 0

\end{pmatrix}x=\begin{pmatrix}(-c/5)+b+(4/5)c\\-b-(4/5)c\\a+b+c\end{pmatrix}

[/tex]



I found that rank = 2. I also found that the vector for the null space of A-transpose is:



[tex]

\begin{pmatrix}1\\1\\1\end{pmatrix}

[/tex]



For part 2, I said that the solution is solvable under the condition that a+b+c = 0 because there is only one row with all zeros.



I am stumped on part 3. I thought that I should just plug a = b = 1, c = -2 into my reduced row echelon, giving me infinitely many solutions. I don't think this is correct though. Any advice would be helpful. Thanks.