Definition of Matrices $a_{ij}, a'_{ij}, b_{ij}$, etc

Question

As stated in the title, what exactly does the following mean?

Let A = $(a_{ij} )$$m×p$ and B = $(b_{ij} )$$p×n$ be two matrices. The product AB is definited to be an $m×n$ matrix whose $(i,j)$-entry is

$a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{ip}b_{pj}$ = $\sum_{k=1}^p a_{ik}b_{kj}$

Now, am i right to say that the above actually meant that for any given $i$ row of A, it is multiplied to the $j$ column of B? If possible, please be specific in using A or $a$, etc and explain this whole idea of $_{ij}$

It would be so appreciated as this whole concept has been bothering me and hindering me on the understanding of further concepts, i.e. the proving of e.g. $ A \begin{pmatrix} B1 & B2 \\ \end{pmatrix} $ = $ \begin{pmatrix} AB1 & AB2 \\ \end{pmatrix} $

Answer 1

I see that you're having trouble understanding the definition of matrix (possibly) and the definition of product of matrices, so I'm going through both of them.

1. What is a matrix? Well, it's pretty much just table of numbers. Something that, in its general form, looks like this: $$\textbf{A} = \begin{pmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \\ a_{m1} & a_{m2} & a_{m3} & & a_{mn} \end{pmatrix}$$ This is an $m \times n$ matrix (means it hat $m$ rows and $n$ columns). What are the $a_{ij}$s? They're called the entries of $\textbf{A}$ and can be elements of an arbitrary field; for our purposes, let's just assume that they are real numbers: we say that $\textbf{A}$ is defined over $\mathbb{R}$. So $a_{ij}$ is just the number located at the $i$-th row and $j$-th column of $\textbf{A}$. The notation $\textbf{A} = (a_{ij})m \times n$ means that $\textbf{A}$ is an $m \times n$ matrix whose entries are $a_{ij}$.

2. Matrix multiplication. Now that we are familiar with the concept of matrix we can go ahead and define the operation of multiplication between matrices, or matrix product. Given two matrixes $\textbf{A} = (a_{ij})m \times p$ and $\textbf{B} = (b_{ij})p \times n$, we define the product $\textbf{AB}$ as an $m \times n$ matrix $\textbf{M} = (m_{ij})$, where $$m_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{ip}b_{pj}$$ This means that $m_{ij}$ is the dot product between the $i$-th row of $\textbf{A}$ and the $j$-th column of $\textbf{B}$. Focus on the fact that, in order for the product $\textbf{AB}$ to exist, it is necessary that the number of columns of $\textbf{A}$ equals the number of rows of $\textbf{B}$ (in the definition, this number is $p$), since otherwise the rows of $\textbf{A}$ and the columns of $\textbf{B}$ would be vectors of different dimensions and thus we wouldn't be able to multiply them (recall that dot product is only defined between vectors of the same dimension).

3. Some examples. Consider matrices $\textbf{A} = \begin{pmatrix} 3 & 0 \\ 2 & -1 \\ 1 & 5 \end{pmatrix}$ and $\textbf{B} = \begin{pmatrix} 4 & 7 \\ -2 & -2 \end{pmatrix}$. Since the number of columns of $\textbf{A}$ ($2$) equals the number of columns of $\textbf{B}$ ($2$), we can compute $\textbf{AB}$ using the definition: $$\textbf{AB} = \begin{pmatrix} 3 \cdot 4 + 0 \cdot (-2) & 3 \cdot 7 + 0 \cdot (-2) \\ 2 \cdot 4 + (-1) \cdot (-2) & 2 \cdot 7 + (-1) \cdot (-2) \\ 1 \cdot 4 + 5 \cdot (-2) & 1 \cdot 7 + 5 \cdot (-2) \end{pmatrix} = \begin{pmatrix} 12 & 21 \\ 10 & 16 \\ -6 & -3 \end{pmatrix}$$ As we expected, it is a $3 \times 2$ matrix. What about $\textbf{BA}$? Tough luck here... Since $\textbf{B}$ has $2$ columns and $\textbf{A}$ has $3$ rows, it doesn't even exist. Now let $\textbf{C} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ and $D = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix}$. Note that both $\textbf{CD}$ and $\textbf{DC}$ are defined! After some calculations (which I will omit because they're not that interesting) one finds $$\textbf{CD} = \begin{pmatrix} 3 & 2 \\ 3 & 2 \end{pmatrix} \quad \text{and} \quad \textbf{DC} = \begin{pmatrix} 2 & 2 \\ 3 & 3 \end{pmatrix}$$

4. Properties of matrix product. The two examples above show that matrix multiplication does not commute, not even when both $\textbf{AB}$ and $\textbf{BA}$ are defined (although it might happen that $\textbf{AB} = \textbf{BA}$, yet it's not the general rule). Instead, two basic properties of matrix product are

   • associativity: for any three matrices $\textbf{A}, \: \textbf{B}, \: \textbf{C}$ with dimensions $m \times p, \: p \times q, \: q \times n$ respectively, it holds $(\textbf{AB})\textbf{C} = \textbf{A}(\textbf{BC})$
   • distributivity over matrix addition: for any three matrices $\textbf{A}, \: \textbf{B}, \: \textbf{C}$ with dimensions $m \times p, \: p \times n, \: p \times n$ respectively, it holds $\textbf{A}(\textbf{B} + \textbf{C}) = \textbf{AB} + \textbf{AC}$; this is the one you mentioned in your OP.

   Proof of these properties is left to you as an exercise for checking your understanding of matrix multiplication.

Hope this helped.