How to multiply matrices (the physicists way) for a total beginner
N.B For this question we are only working in in flat cartesian space, not curved space-time.
Question: Consider $A$, $B$, $C$ and $D$ to be $n \times n$ matrices. Write down the following matrix multiplications using index notation and <PERSON> summation rules: 1. $A=B(C+D)$ 2. $A=BCD$
According to my notes:
$$C=AB \qquad \text{means}\qquad C^i{\,j}=A^i{\,_k}B^k{\,_j}\tag{1}$$ $$D=BA \qquad \text{means}\qquad D^i{\,_j}=B^i{\,_k}A^k{\,_j}=A^k{\,_j}B^i{\,_k}\tag{2}$$ $$E=A^TB \,\,\,\quad \text{means}\qquad E^i{\,_j}=A{{\color{red}{k}}}^{\,\color{blue}{i}}B^k{\,_j}\tag{3}$$
For question 1. multiplying out gives $A =BC+BD$, and following the rules above I think the answer in tensor form should be $A^i{\,_j}=B^i{\,_k}C^k{\,_j}+B^i{\,_k}D^k{\,_j}$
For question 2. 490
by my logic, $A=BCD$, so in tensor form this should be $A^i{\,j}=B^i{\,_k}C^k{\,\ell}D^\ell{\,_j}$
However, according to the authors' solutions, both of my answers are wrong:
$A=B(C+D)\implies A_{ij}=B_{ik}C^k{\,j}+B{ik}D^k{\,j}=\delta^{k\ell}B{ik}\left(C_{\ell j}+D_{\ell j}\right)$ $A=BCD \implies A_{ij}=B_{ik}C^k{\,\ell}D^\ell{\,_j}=\delta^{km}\delta^{\ell n}B{ik}C_{m\ell}D_{nj}$
So I followed the rules given by $(1)$, $(2)$, and $(3)$ and got the wrong answer. But, before I ask about the authors' solutions, there is something I don't understand regarding equation $(3)$.
According to a small section of my notes, I have that:
From the notes above, I think equation $(3)$ should be $$E^i{\,j}=A{{\color{red}{i}}}^{\,\color{blue}{k}}B^k{\,_j}\tag{4}$$ (where I have swapped the indices for the red and blue parts), since the transpose operation interchanges rows and columns and the section of notes shown above states that the lower index of the matrix labels the columns. So why is equation $(3)$ written that way when rows and columns are clearly being interchanged? By the way, I know that equation $(4)$ which I have written is actually wrong also, since the 'dummy index', $k$ is written up twice. But, what I am questioning here is what the transpose does to the indices.
From the image above, I have also been taught how to use the <PERSON> delta metric (or identity matrix), which in the case of a $2\times 2$ identity matrix is $\delta^{ij}=\delta_{ij}=\begin{pmatrix}1 & 0 \ 0 & 1 \\end{pmatrix}$. So the way I see it, multiplying by the $\delta^{ij}$ essentially changes a row vector to a column vector, and my understanding of this is because the 'dummy index' ($j$ in the example in the top right of the image) is contracted upon so that the <PERSON> is only non-zero when $i=j$ and this 'somehow' changes the original $U_i$ to $U^i$. Why this works the way it does is a question for another time, but for now, I would like to focus on trying to understand why the authors' solutions look the way they do for matrix multiplication. 1021