Using Suffix Notation, I have to show that the $n \times n$ identity matrix is commutative with any $n \times n$ martix with respect to matrix multiplication.
We have just been introduced to the Kronecker delta symbol and so I was hoping someone could tell me if the following is a correct way to go about the problem:
Let $M$ be any $n \times n$ matrix. We know that $\delta_{ij}$ is the suffix notation of the $n \times n$ identity matrix.
$M_{ij}\delta_{jk} = M_{ik} = \delta_{ij}M_{jk}$
Hence, the $n \times n$ identity matrix is commutative with any $n \times n$ martix with respect to matrix multiplication.
I'd appreciate any help! Thanks!
Maybe it would be better to write summations:
$$(MI)_{ik}=\sum_{j=1}^n M_{ij}\delta_{jk}=M_{ik}\delta_{kk}+\sum_{j\neq k} M_{ij}\delta_{jk}=M_{ik},$$
and similarly for the right-handside. Implicitly you are using the fact that two matrices are equal (by definition) if all of their matching entries are equal.