[Math] Find values of a and b that make the system consistent

linear algebramatrices

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?

Best Answer

"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.

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