Matrix space/K^2/Multiplication with fixed matrix/Exercise

We consider the space of matrices ${}\operatorname {Mat} _{2}(K)$, and we fix the matrix ${}A={\begin{pmatrix}9&-8\\-7&6\end{pmatrix}}$.

1. Show that the mapping

   $\varphi \colon \operatorname {Mat} _{2}(K)\longrightarrow \operatorname {Mat} _{2}(K),M\longmapsto A\circ M,$

   is linear.

2. Determine the describing matrix of ${}\varphi$ with respect to a suitable basis.
3. Determine the kernel and the image of ${}\varphi$.

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