Here's my system of linear equations:
$\begin{cases}
x + 2y + 2z = 1\\x + ay + 3z = 3\\x + 11y +az = 0\\
\end{cases}$
Thus I have the augmented matrix $\left[\begin{array}{ccc|c}1&2&2&1\\1&a&3&3\\1&11&a&0\end{array}\right]$
By row reduction, I obtain:
$\left[\begin{array}
{ccc|c}1&2&2&1\\0&a-2&1&2\\0&9&a-2&-1\end{array}\right]$
Unfortunately, I am stuck at this stage. I have tried swapping rows around but I didn't have much luck.
Update: I have managed to solve this with the use of the determinant.
Matrix of minors:
$\left[\begin{array}
{ccc}a^2-33&a-3&11-a\\2a-22&a-2&9\\6-2a&1&a-2\end{array}\right]$
Matrix of cofactors:
$\left[\begin{array}
{ccc}a^2-33&3-a&11-a\\22-a&9&a-2\\6-2a&-1&a-2\end{array}\right]$
Adjugate matrix:
$\left[\begin{array}
{ccc}a^2-33&22-2a&6-2a\\3-a&a-2&-1\\11-a&-9&a-2\end{array}\right]$
Det(A) = $1(a^2 – 33) + 2(3 – a) + 2(11 – a) = a^2 – 4a – 5$
$(a – 5)(a + 1) = 0$
Thank you all for your help!
Best Answer
Let's rearrange the augmented matrix representing the system of linear equations, by swapping $R_2$ and $R_3$ (to avoid division by $a-2$ while row-reduction), to get
$\left[\begin{array}{ccc|c}1&2&2&1\\1&11&a&0\\1&a&3&3\end{array}\right]$ and a subsequent row-reduction will give you
$\left[\begin{array}{ccc|c}1&2&2&1\\0&9&a-2&-1\\0&0&1-\frac{(a-2)^2}{9}&2+\frac{a-2}{9}\end{array}\right]$
Unique solution: in order for the system to have a unique solution, the coefficient and the augmented matrix both must be of full rank i.e., both must have rank $n=3$. To make that happen, we must have $1\neq\frac{(a-2)^2}{9} \implies a \notin \{-1,5\}$.
No solution: for this to happen, we need to have the augment matrix rank higher than the coefficient matrix rank. This can happen if the coefficient matrix has rank < 3, e.g., if $1=\frac{(a-2)^2}{9} \implies a \in \{-1,5\}$ and augmented matrix has rank $n=3$ simultaneously, which means $2+\frac{a-2}{9} \neq 0 \implies a \neq -16$, i.e., for $a=-1,5$ the system will have no solution.
Infinitely many solutions: it can happen iff both the coefficient and the augmented matrix are rank-deficient, i.e. for both of them the rank is $<3$. For this the last row of the augmented matrix needs to be zero. Which implies both $1-\frac{(a-2)^2}{9}=0 \implies a \in \{-1,5\}$ and $2+\frac{a-2}{9}=0 \implies a=-16$ simultaneously. Which is clearly impossible.