Let A be the matrix , where no one of a,b,c,d is zero. It is required to find a non-zero 2x2 matrix X such that AX+XA=0, where 0 is the zero 2x2 matrix. Prove that either
(a) a+d=0, in which case the general solution for X depends on two parameters, or
(b) ad-bc=0, in which case the general solution for X depends on one parameter.
I don't know where to begin other than naming X= and then find the matrices AX and XA.
You have to work harder and make a bigger effort. The matrix of coefficients of the system ( in the unknonws ) is:
. Divide now the first row by (why can you?) and substract from each of the 2nd, 3rd rows a corresponding multiple of the first one to obtain:
. Interchange now the 2nd and 4th rows, divide the new 2nd row by (again, why can you) and ...etc. This is
just the Gauss-Jordan reduction method of a matrix to echelon form!:
...or something like this (check this carefully: I didn't!) ...and etc.