Showing an endomorphism is not surjective Let $$A=\begin{pmatrix}2&-2\\2&-2\end{pmatrix}$$ and the endomorphism $f_A:M_2(\mathbb R)\longrightarrow M_2(\mathbb R); B\longmapsto AB$. I want to show that $f_A$ is not surjective.
My try: $\ker f_A$ is clearly shown to be containing elements other than the null matrix, so $f_A$ is not injective. Since this is endomorphism in finite dimension then $f_A$ is not injective if and only if $f_A$ is not surjective.
Is there any other way to show that $f_A$ is not surjective?

Your approach works (it'd be better to explicitly write down a nonzero element of $\mathrm{ker} \, f_A$). Alternatively, every element of the image satisfies $\det(f_A(B)) = \det(AB) = 0 \cdot \det(B) = 0$. Of course there are matrices with nonzero determinant, and they are therefore not in the image.
