<s>[INST] How many matrices satisfy this equality? How many matrices $A\in\mathcal{M}_{3\times 3} (\mathbb{N})$ satisfy this equality?
$$\begin{pmatrix}
1 \ \ 2 \ \ 4
\end{pmatrix}\cdot A=\begin{pmatrix}
3 \ \ 2 \ \ 1
\end{pmatrix}$$
I tried with examples and I found just one but I want to know how to approach this exercise.The right answer is $3$
 [/INST] Let $A=[x_{i,j}]$ Then we get
$$x_{1,1} +2 x_{2,1} +4 x_{3,1} = 3$$
$$ x_{1,2} + 2x_{2,2} + 4 x_{3,2} = 2 $$
$$x_{1,3} + 2 x_{2,3} + 4 x_{3,3} = 1$$
We shall count the number of solutions for each equation.
First equation has 2 solutions:
Since $4>3$ we have that $x_{3,1}=0$. Similarly $x_{2,1}<2$. If $x_{2,1}=1$ then $x_{1,1}=1$ and if $x_{2,1}=0$ then $x_{1,1}=3$. Therefore there are two solutions for the first equation.
Second equation has 2 solutions:
As before, $x_{3,2}=0$. Also $x_{2,2}$ is at most $1$, hence if $x_{2,2}=1$ then $x_{1,2}=0$ and if not then $x_{2,2}=0$ and $x_{1,2}=1$.
Third equation has 1 solution:
$4,2>1$ so $x_{2,3},x_{3,3}=0$ and so the only solution is $x_{1,3}=1$.
Therefore there is a total of $4$ solutions for all of the equations.
$$A_1 =\begin{bmatrix} 3 & 0 &1  \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{bmatrix}, A_2=\begin{bmatrix} 1 & 0 &1  \\ 1 & 1 & 0 \\ 0 & 0 & 0\end{bmatrix}$$
$$A_3 =\begin{bmatrix} 3 & 2 &1  \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}, A_4=\begin{bmatrix} 1 & 2 &1  \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$$
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