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Is the identity matrix \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} and its scalar multiples always commutative?

I would greatly appreciate it if people could please clarify this.

2 Answers 2

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The identity matrix commutes with all matrices since $IM = MI = M$. Now, numbers can be shifted around as you please, so that if $K = rI$, we have that $KM - MK = r(IM - MI) = 0$.

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    And multiples of the identity matrix are also commutative? – 2017-02-21
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    Every matrix is a multiple of the identity matrix, so no. – 2017-02-21
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    @AJY I think it means a scalar multiple, in which case yes – 2017-02-21
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    @Daminark Yes, I meant scalar multiples of the identity matrix. Thanks. – 2017-02-21
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Yes, and you can write any matrix as an addition of elementary matrices.