The homogeneous matrix equation involves a zero right-hand side.
This equation has a nonzero solution if the matrix is singular. To test if a matrix is singular, you can compute the determinant.
To find the solution of the homogeneous equation you can use the function NullSpace. This returns a set of orthogonal vectors, each of which solves the homogeneous equation. In the example below, there is only one vector.
This demonstrates that the solution in fact solves the homogeneous equation.
The function Chop can be used to replace approximate numbers close to 0.
The solution to the homogeneous equation can be used to form an infinite number of solutions to the inhomogeneous equation. This solves an inhomogeneous equation.
The solution does in fact solve the equation.
If you add to sol an arbitrary factor times the homogeneous solution, this new vector also solves the matrix equation.