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Matrices that commute with all matrices

Please! Can someone explain to me with details why the a coefficient is equal to zero when $k≠i$.

F. A. Mala
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jakehoffman
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1 Answers1

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Give it a try in a simple $2\times 2$ case. If a matrix $A=\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}$ commutes with all matrices, it then must commute with a matrix such as $E_{12}=\begin{bmatrix}0&1\\0&0\end{bmatrix}$. However:

$$AE_{12}=\begin{bmatrix}0&a_{11}\\0&a_{21}\end{bmatrix}$$ $$E_{12}A=\begin{bmatrix}a_{21}&a_{22}\\0&0\end{bmatrix}$$

which directly tells you that $a_{21}=0$. Similarly, you would use $E_{ij}$ (the matrix where $ij$'th entry is $1$ and all the other entries are $0$), with $i\ne j$, to prove $A_{ji}=0$.