[Math] Finding multiple solution of a matrix

linear algebramatrices

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:

  1. a unique solution
  2. a one parameter solution
  3. a two parameter solution
  4. 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: $\{(t,t,u):t,u \in \mathbb{R}\}$.

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