Consider the following vector $\vec x$:
$$\vec x= \vec e_a x^a $$
$$x^a= \left( \begin{matrix} A \\ D \\ G \\ \end{matrix} \right) $$
$$x^1= A, x^2=D,x^3=G $$
$$\vec e_1= \hat i ,\vec e_2=\hat j, \vec e_3=\hat k $$
$$\vec e_1= \left( \begin{matrix} \hat i & 0 & 0 \\ \end{matrix} \right) = \left( \begin{matrix} 1 & 0 & 0 \\ \end{matrix} \right) \left( \begin{matrix} \hat i \\ \end{matrix} \right) = e_1 \hat i $$
$$\vec e_2= \left( \begin{matrix} 0 & \hat j & 0 \\ \end{matrix} \right) = \left( \begin{matrix} 0 & 1 & 0 \\ \end{matrix} \right) \left( \begin{matrix} \hat j \\ \end{matrix} \right) = e_2 \hat j $$
$$\vec e_3= \left( \begin{matrix} 0 & 0 & \hat k \\ \end{matrix} \right) = \left( \begin{matrix} 0 & 0 & 1 \\ \end{matrix} \right) \left( \begin{matrix} \hat k \\ \end{matrix} \right) = e_3 \hat k $$
$$ \vec x= \vec e_1 A+ \vec e_2 D + \vec e_3 G \quad or \quad \vec x= \left(\vec e_1 A+ \vec e_2 D + \vec e_3 G \right)_{1x1} $$
$$ \vec x= \vec e_1 x^1+ \vec e_2 x^2 + \vec e_3 x^3 $$
$$ \vec x= \left( \begin{matrix} \vec e_1 & \vec e_2 & \vec e_3 \\ \end{matrix} \right)_{1x3} \left( \begin{matrix} x^1 \\ x^2 \\ x^3 \\ \end{matrix} \right)_{3x1} $$
$$\vec x= \left( \begin{matrix} e_1 \hat i & e_2 \hat j & e_3 \hat k \\ \end{matrix} \right)_{1x3} \left( \begin{matrix} x^1 \\ x^2 \\ x^3 \\ \end{matrix} \right)_{3x1}$$
$$\vec x= \left( \begin{matrix} e_1 & e_2 & e_3 \\ \end{matrix} \right)_{1x3} \left(\begin{matrix} \hat i & 0 & 0 \\ 0 & \hat j & 0 \\ 0 & 0 & \hat k \\ \end{matrix} \right)_{3x3} \left( \begin{matrix} x^1 \\ x^2 \\ x^3 \\ \end{matrix} \right)_{3x1}$$
Now I am going to say my opinion about the above. Please do correct me if I am wrong in this paragraph. In the above, the 3x3 matrix is a tensor of rank 3.But the 1x3 & 3x1 matrices are matrices of column vector- & row vector- components respectively. Vector is a tensor of rank 1. Vector components are usually written as a matrix of only 1 column (or as a column matrix of scalars). These components are contravariant.Co-vector components are usually written as a matrix of only 1 row (or as a row matrix of scalars).These components are co-variant.The basis vectors of a vector are co-variant. A matrix of vector components $\neq$ vector. ie $ x^a \neq \vec x $ But the matrix product of matrix with vector components ($x^a$) & matrix with basis vectors ($\vec e_a \quad or \quad (\vec e_1 \quad \vec e_2 \quad \vec e_3)$) is a vector. The matrix with basis vectors $( \vec e_1 \quad \vec e_2 \quad \vec e_3)$ is not a vector. But it is just the matrix product of the matrix with basis vector components $(e_1 \quad e_2 \quad e_3)$ & the tensor $\left(\begin{matrix} \hat i & 0 & 0\\ 0 & \hat j & 0 \\ 0 & 0& \hat k\\ \end{matrix}\right)$. And a matrix product is a matrix too.
Now, consider the following matrix where all its components are scalars:
$$ Q_M=\left( \begin{matrix} A & B & C\\ D & E & F\\ G & H & I\\ \end{matrix} \right) $$
And consider the following 3x3 matrix with components which are vectors:
$$ \left( \begin{matrix} A \\ B\\ C\\ \end{matrix} \right) \left(\begin{matrix} \hat i & \hat j & \hat k \\ \end{matrix} \right)= \left( \begin{matrix} A\hat i & A\hat j & A\hat k\\ B\hat i & B\hat j & B\hat k\\ C\hat i & C\hat j & C\hat k\\ \end{matrix} \right)_{3x3} = Q_{T1}$$
Also consider the following matrix with components which are vectors:
$$ Q_{T2}= \left( \begin{matrix} A\hat i & B\hat j & C\hat k\\ D\hat i & E\hat j & F\hat k\\ G\hat i & H\hat j & I\hat k\\ \end{matrix} \right) $$
Is it correct if I say $Q_M$ is just a matrix(& not a tensor) but $Q_{T1}$ & $Q_{T2}$ are tensors? And the reason why I say so is because $Q_M$ has components that are purely scalars but $Q_{T1}$ & $Q_{T2}$ have components with different magnitude & direction. Its similar to saying $x^a$ is not a vector but $\vec x$ is a vector & that $x^a$ is just a column matrix of vector components. If a matrix couldn't be written as a matrix product of column matrix (of contravariant components) & row matrix (of covariant components) (eg- matrix of basis vectors) ,then is it not then a tensor? Is it also correct if I believe not all matrices are tensors ,not all tensors are matrices ,not all arrays are matrices (a matrix is a 2 dimensional array),all matrices are arrays ,not all arrays are tensors ,all tensors are arrays, not all sets are arrays & all arrays are sets?