If a matrix appears as a variable in a polynomial equation then we can solve for it using techniquees similar to those used in ordinary algebra. In particular, we can factorise. If the polynomial in the matrix equals zero, then each factor equals zero and we can solve for the matrix. It should be noted though, that though the method below find solutions for the matrix, it does not find all solutions.
Example: Solve the matrix equation
Write the equation as
and factorise
then
or
Example: Solve the matrix equation
Write the equation as
and factorise
then
or
To see why this method does not find all solutions, consider the first equation
If
then