Augmented matrix linear algebra

[devilaudy]

Homework Statement

Given the system whose augmented matrix is
 1 1 1 1 
 1 −1 0 a 
 0 1 b 0 
Determine (if possible) conditions on a and b such that this system has (a) no solution (b) many solutions (c) a unique solution.

Homework Equations

-Row reduction
-No solution: row of zero
-Many solutions: one or more free variables
-Unique solution: pivot in every column

The Attempt at a Solution

I tried to do it, but my answers were wrong. The good answers are a) b=1/2, a≠1 b) b=1/2, a=1 c) b≠1/2.

I tried to reduce the matrix and I had:

 0 0 1-2b 1-3a 
 1 0 b 2a 
 0 1 b a 

Can someone explain how to reduce the matrix properly or what am I doing wrong?

 

[Mark44]

Homework Statement

Given the system whose augmented matrix is
 1 1 1 1 
 1 −1 0 a 
 0 1 b 0 
Determine (if possible) conditions on a and b such that this system has (a) no solution (b) many solutions (c) a unique solution.

Homework Equations

-Row reduction
-No solution: row of zero
-Many solutions: one or more free variables
-Unique solution: pivot in every column

The Attempt at a Solution

I tried to do it, but my answers were wrong. The good answers are a) b=1/2, a≠1 b) b=1/2, a=1 c) b≠1/2.

I tried to reduce the matrix and I had:

 0 0 1-2b 1-3a 
 1 0 b 2a 
 0 1 b a 

Can someone explain how to reduce the matrix properly or what am I doing wrong?

Right off the bat I swapped the 2nd and 3rd rows. After that, I row reduced to get a matrix in echelon form. I get the same answers as your "good" ones.