Find values of a and b that make the system consistent In the augmented matrix:
$$\left(\begin{array}{rrr|r}
1 &-2 &4 & 7\\
0 &a^2 - 1& a & 3\\
0 &0 &b & -3
\end{array}\right).$$
How do I determine values for $a$ and $b$ that make the system consistent?
 A: "What values of $a$ and $b$ make the system consistent?" asks for all pairs of values for $a$ and $b$ that make the system consistent, not just a single one.
To find them, do Gaussian elimination, leaving expressions that involve $a$ and $b$ indicated. Be particularly careful when dividing by an expression involving $a$ or $b$, since you must ensure that the expression you divide by is nonzero.
For example, a first step, since the matrix is already in upper triangular form, might be to divide the last row by $b$ to make the $(3,3)$ entry into a $1$; but in order to "divide by $b$", you need $b\neq 0$. What happens in $b=0$ to the system? You need to consider that. Then, if you assume $b\neq 0$, then you can divide by $b$ and proceed from there.
A: I think I have come up with an answer, I would appreciate it if it could be verified.
I observe that b must be non-zero for a consistent system. a can be any real number. If a = 1, then b must equal -1. If a = -1, then b must equal 1. There is a unique solution given these conditions where a != 1 and a != -1. There are infinitely many solutions where a = 1 and b = -1, or when a = -1 and b = 1.
A: I want to make sure I have the hang of this with one more example.
Given the matrix:
1 0 2 5 | 2
0 c c 0 | 1
0 0 c 0 | c
0 0 0 cd | c + d
I observe that c must be non-zero and that d must be non-zero. Given these conditions there is always a unique solution. Is this correct?
