I took a course in applied matrix methods recently, and we often solved "matrix equations" such as
$ A+BX = C$
By using the information given about their invertability or non-zeroness, and multiplying by the appropriate matrices on either side
So in my mind, it was okay to take the above equation and write $$C(A+BX) = C^2$$ and $$ C(A+BX)C = C^3$$
However, I have run into a problem that has led me to believe more caution should be exercised with such manipulations, it is as follows: Let A be a real n×n matrices such that $$ A^3 = 0$$ and suppose $$X + AX + XA^2 = A$$ My natural first approach was to split it into cases, with $A=0$ and $A \neq 0, A^2 =0$ for the second case however, I thought to multiply both sides of the equation by A, both pre and post multiplication, and go from there, you then get $$AXA = 0$$, which has solutions $cI$ for scalars c, but actually trying an example, I noted that this X was not a solution, leaving me quite confused. What went wrong here?
Your matrix is singular, so multiplying by it is akin to multiplying by zero in scalar equations - you get a condition that is necessary, but not sufficient.The sledgehammer approach is to use the Jordan normal form for $A,$ though multiplying by $A$ obviously does give you some information about $X.$