It is well-known that for the general linear system
$$\mathbf{X}\mathbf{a}=\mathbf{y}$$
where $\mathbf{X}$ is a rectangular matrix and $\mathbf{a}$ and $\mathbf{y}$ are column vectors of compatible sizes, the solution given by the Moore-Penrose pseudoinverse $\mathbf{a}=\mathbf{X}^+\mathbf{y}$ always exist and
if $\mathbf{y}$ is in the column space of $\mathbf{X}$ (i.e. the system is satisfiable), $\mathbf{a}=\mathbf{X}^+\mathbf{y}$ is the solution with minimum norm.
if $\mathbf{y}$ is not in the column space of $\mathbf{X}$ (i.e. the equality is not satisfiable), then $\mathbf{a}=\mathbf{X}^+\mathbf{y}$ is the best approximation for the equality, i.e. $\mathbf{a}=\mathbf{X}^+\mathbf{y} = \text{argmin}_\mathbf{a} \|\mathbf{Xa}-\mathbf{y}\|^2$
furthermore if $\mathbf{X}$ has linearly-independent columns then $\mathbf{a}=(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{y}$
Now regarding the Matrix Equation
$$\mathbf{XA}=\mathbf{Y}$$ where $\mathbf{A}$ is the unknown Matrix and $\mathbf{X}$ and $\mathbf{Y}$ are matrices of compatible sizes
Can we also say that $\mathbf{A}=\mathbf{X}^+\mathbf{Y}$ has the same properties as before? Namely
if each column of $\mathbf{Y}$ belongs to the column space of $\mathbf{X}$ then $\mathbf{A}=\mathbf{X}^+\mathbf{Y}$ is the solution with minimum (matrix) norm?
if some column of $\mathbf{Y}$ does not belong to the column space of $\mathbf{X}$ then $\mathbf{A}=\mathbf{X}^+\mathbf{Y}$ is the best approximation for the system, i.e. $\mathbf{A}=\mathbf{X}^+\mathbf{Y}=\text{argmin}_\mathbf{A}\|\mathbf{XA}-\mathbf{Y}\|^2$ for some matrix norm
Furthermore if $\mathbf{X}$ has linearly-independent columns then $\mathbf{A}=(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{Y}$?