Question: Let \(A\in {{\mathbb{C}}^{m\times n}},B\in {{\mathbb{C}}^{n\times q}},C\in {{\mathbb{C}}^{m\times n}},D\in {{\mathbb{C}}^{n\times q}}\)

(a) When is the matrix equation \(AX=C\) solvable for \(X\in {{C}^{n\times n}}\). When is the solution unique?

(b) When is the matrix equation \(XB=C\) solvable for \(X\in {{C}^{n\times n}}\). When is the solution unique?

(c) Under what conditions does \(TA=TC\) imply that \(A=C\).