<s>[INST] Find a invertible matrix $Q$ such that $AQ$ = $B$ Hi I have calculate this matrices:
$$A = \begin{pmatrix} 1 & 0 & \frac{1}{3}\\ 0 & 0 & -\frac{1}{3} \\ 1 & 1 & \frac{2}{3}\end{pmatrix}$$
$$B = \begin{pmatrix} \frac{4}{3} & \frac{1}{3} & 1\\ -\frac{1}{3} & -\frac{1}{3} & 0 \\ \frac{5}{3} & \frac{8}{3} & 2\end{pmatrix}$$
And I'm trying to find a invertible matrix $Q$ such that $AQ$ = $B$, but I'm stuck.
Can you help me?
I have not seen determinats yet
 [/INST] Suppose that we have $$A = \begin{pmatrix} 1 & 0 & \frac{1}{3}\\ 0 & 0 & -\frac{1}{3} \\ 1 & 1 & \frac{2}{3}\end{pmatrix}$$
$$B = \begin{pmatrix} \frac{4}{3} & \frac{1}{3} & 1\\ -\frac{1}{3} & -\frac{1}{3} & 0 \\ \frac{5}{3} & \frac{8}{3} & 2\end{pmatrix}$$
and we want $Q$ such that $AQ =B$. Let
$$ Q = \begin{pmatrix} a & b &c \\  d & e & f \\ g & h & i \end{pmatrix}$$
Then using matrix multiplication you'd see that
$$ AQ =  \begin{pmatrix} 1 & 0 & \frac{1}{3}\\ 0 & 0 & -\frac{1}{3} \\ 1 & 1 & \frac{2}{3}\end{pmatrix}   \begin{pmatrix} a & b &c \\  d & e & f \\ g & h & i \end{pmatrix} = \begin{pmatrix} a + \frac{g}{3}  & b + \frac{h}{3} & c + \frac{i}{3} \\ -\frac{g}{3} & \frac{-h}{3} & \frac{-i}{3} \\ a + d + \frac{2g}{3} & b + e + \frac{2h}{3} & c + f + \frac{2i}{3} \end{pmatrix} $$
now set it equal to $B$
$$ \begin{pmatrix} a + \frac{g}{3}  & b + \frac{h}{3} & c + \frac{i}{3} \\ -\frac{g}{3} & \frac{-h}{3} & \frac{-i}{3} \\ a + d + \frac{2g}{3} & b + e + \frac{2h}{3} & c + f + \frac{2i}{3} \end{pmatrix} =  \begin{pmatrix} \frac{4}{3} & \frac{1}{3} & 1\\ -\frac{1}{3} & -\frac{1}{3} & 0 \\ \frac{5}{3} & \frac{8}{3} & 2\end{pmatrix}$$
Multiply by $3$ on both sides and we get
$$ \begin{pmatrix} 3a + g  & 3b + h & 3c + i \\ -g & -h & -i \\ 3a + 3d + 2g & 3b + 3e 2h & 3c + 3f + 2i \end{pmatrix} =  \begin{pmatrix} 4 & 1 & 3 \\ -1 & -1 & 0 \\ 5 & 8 & 6\end{pmatrix}$$
Then you directly get $g,h,i$
$$ -g = -1 \implies g = 1 \\ -h = -1 \implies h = 1 \\ -i = 0 \implies i=0$$
Now substitute those into the top row
$$ 3a +1 =  4 \implies a =1 \\ 3b + 1 = 1 \implies b = 0 \\ 3c = 3 \implies c =1 $$
then continue and get the last three
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