<s>[INST] Solution to linear equations as parameterized matrices. I want to solve the following matrix equation:
$$
\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} 3 & 2 \\ 4 & 2 \end{pmatrix}=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}
$$
and I thus get for equations:
$$
\\2a +4b -2c = 0
\\2a + b -2d= 0
\\-3a -c +4d = 0
\\-3b +2c -2d = 0
$$
which reduces to:
$$
\\a+2b-c=0
\\3b-2c+2d=0
$$
Now I want to parameterize this with parameters $s$ and $t$, and intuition tells me I should be able to express the solution as:
$$
T=\begin{pmatrix} a & b \\ c & d \end{pmatrix}=s\begin{pmatrix} x & x \\ x & x \end{pmatrix}+t\begin{pmatrix} x & x \\ x & x \end{pmatrix}
$$
(where $x_i$ is som natural number)
This should be similar to parameters with vectors, but I havn't been able to solve this.
So whats the trick here?
Thanks!
/Alexander
 [/INST] Choose two variables to represent $s$ and $t$. I'll use $a=s$ and $b=t$.
$$\begin{align}c&=s+2t\\
\\
2d & = 2c-3t \\
2d & = 2(s+2t)-3t \\
2d & = 2s + t \\
d &= s + \frac{1}{2}t
\end{align}
$$
$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}=\begin{bmatrix} s & t \\ (s+2t) & (s+\frac{1}{2}t) \end{bmatrix}=\begin{bmatrix} s & 0 \\ s & s \end{bmatrix}+\begin{bmatrix} 0 & t \\ 2t & \frac{1}{2}t \end{bmatrix}$$
</s>