I have this matrix from a book's exercise.
$$
\left[
\begin{array}{@{}cccc@{}}
a&0&b & 2 \\
a& a& 4 & 4 \\
0&a& 2 & b\\
\end{array}
\right]
$$
be the augmented matrix for linear system. Find for what values of ${a}$ and ${b}$ the system has:
- a unique solution
- a one parameter solution
- a two parameter solution
- no solution
My questions:
- How do i solve this? what should i use? gauss-jordan elimination? row operation?
crammer's rule? - Can someone please explain what one parameter means? i have hard time comprehending it
Please help.
Thanks.
Best Answer
I like to think of $a$ and $b$ as being real numbers I don't happen to know.
Your task is to write something like "There is a unique solution if and only if [some conditions on $a$ and $b$]" and so on for the other cases.
You can determine the number of solutions from the row echelon form (found via Gaussian elimination, i.e., row operations).
It might not be consistent (no solutions). This occurs if and only if there is a row such as $[0\ 0\ 0\ 1]$ in the row echelon form (which is equivalent to the equation $0=1$).
There might be a unique solution. In this case, there will be three leading entries (one in each row) in row echelon form.
There might be a one-parameter solution. In this case, there will be two leading entries (and thus a row of zeroes) in row echelon form.
And so on.
A "one-parameter solution" is when the solution space is a line. As a more simple example $$ \left( \begin{array}{cc|c} 1 & -1 & 0 \\ 2 & -2 & 0 \\ \end{array} \right) $$ has infinitely many solutions: $(x,y)=(t,t)$ for all $t \in \mathbb{R}$. Here we have one parameter: $t$. The solution space is the line $\{(t,t):t \in \mathbb{R}\}$.
The system of equations $$ \left( \begin{array}{ccc|c} 1 & -1 & 0 & 0 \\ 2 & -2 & 0 & 0 \\ \end{array} \right) $$ would have a two-parameter solution space: $\{(t,t,u):t,u \in \mathbb{R}\}$.