Let A be a square matrix of order n, $\phi$ a linear form on the set of square matrices of order n with real entries and $f:M\mapsto M-\phi(M)A$ a function. Find a necessary and sufficient condition on $\phi(A)$ for $f$ to be a bijective function.
Since $f$ is a linear map on a finite dimension space, I tried finding conditions on when it is an injection but it came to nothing.
Note that $$ f(M) = 0 \implies M - \phi(M)A = 0 \implies M = \phi(M)A $$ That is, the map $f$ has a non-trivial kernel if and only if there exists a non-zero matrix $M$ such that $M = \phi(M)A$.
However, in order to have $M$ be a multiple of $A$, we must have $M = \alpha A$ for some $\alpha$. Thus, we can rewrite the above as $\alpha A= \phi(\alpha A)A = \phi(A)\alpha A$.
Putting it all together, we see that $f$ fails to be bijective if and only if $\phi(A) = 1$.