## Question

###### Let $B$ be a fixed matrix in $M_{n}(\mathbb{F})$, where $\mathbb{F}$ is an arbitrary field. Define $T: M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})$ by $$ T(A)=A B-B A . $$ Prove that $T$ is a linear operator on $M_{n}(\mathbb{F}) .$ Describe the kernel of $T$ if $B$ is scalar matrix. Find the range of $T$ in case $B$ is a diagonal matrix.

Let $B$ be a fixed matrix in $M_{n}(\mathbb{F})$, where $\mathbb{F}$ is an arbitrary field. Define $T: M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})$ by $$ T(A)=A B-B A . $$ Prove that $T$ is a linear operator on $M_{n}(\mathbb{F}) .$ Describe the kernel of $T$ if $B$ is scalar matrix. Find the range of $T$ in case $B$ is a diagonal matrix.

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