Whenever we see a matrix $A=\bigl( \begin{smallmatrix} 3 & 2 \\ 1 & 2 \end{smallmatrix} \bigr)$ and $v=(3, 2),$ we can visualize that $(3, 2)$ represent the coordinates of $\mathbf i$ vector and $(1,2)$ represents the $\mathbf j$ vector. And to visualize $A$ as a whole, we can see it as a matrix that transforms can the vector $v.$ when we multiply it.

So, we can visualize $A^{-1}$ as a matrix that *undoes* the transformation by $A.$ Similarly, the determinant of $A$ gives the area.

How can I visualize the transpose of a matrix as well as matrices that have a complex number as an element? Thank you.