Alignment across multiple equations

Question

Right now, my document looks like this:

Or, like this:

It bothers me a lot that the equations are scattered all over the place and are not evenly aligned. I want, for example, each line be aligned on the left side with Problem 1 or Problem 2. Any suggestions?

\documentclass[paper=a4, fontsize=11pt]{scrartcl} % A4 paper an
\usepackage{amsmath,amsthm,amssymb}
\usepackage{enumitem}
\begin{document}

\begin{enumerate}[label=\bfseries Problem \arabic*:]

\item %PROBLEM 1

Find $\textbf{A}+\textbf{B}$, $\textbf{AB}$, $\textbf{BA}$.\\\\


a) $\textbf{A} = \begin{bmatrix}  1 & 2  \\   3  & -1  \end{bmatrix}\,\,\,\,\,$ and $\,\,\,\,\,\textbf{B} = \begin{bmatrix} 2 & 1  \\1  & 1 \end{bmatrix}$. \,\,\,\,\,\textbf{A} is $2\times2$ and \textbf{B} is $2\times2$.

 \[\textbf{A}+\textbf{B}=\begin{bmatrix}  1 & 2  \\ 3  & -1\end{bmatrix} + \begin{bmatrix} 2 & 1  \\1  & 1 \end{bmatrix}=\begin{bmatrix} 1+2 & 2+1  \\3+1  & -1+1 \end{bmatrix}=\begin{bmatrix} 3 & 3  \\4  & 0 \end{bmatrix}\]

 \[\textbf{AB}=\begin{bmatrix}  1 & 2  \\ 3  & -1\end{bmatrix} \cdot \begin{bmatrix} 2 & 1  \\1  & 1 \end{bmatrix}=\begin{bmatrix} 1\cdot2+2\cdot1 & 1\cdot 1+2\cdot1  \\3\cdot2+(-1)\cdot1  & 3\cdot1+(-1)\cdot1 \end{bmatrix}=\begin{bmatrix} 4 & 3  \\5  & 2 \end{bmatrix}\]

 \[\textbf{BA}= \begin{bmatrix} 2 & 1  \\1  & 1 \end{bmatrix}\cdot \begin{bmatrix}  1 & 2  \\ 3  & -1\end{bmatrix}=\begin{bmatrix} 2\cdot1+1\cdot3 & 2\cdot 2+1\cdot(-1)  \\1\cdot1+1\cdot3  & 1\cdot2+1\cdot(-1) \end{bmatrix}=\begin{bmatrix} 5 & 3  \\4  & 1 \end{bmatrix}\]\\




b) $\textbf{A} = \begin{bmatrix} 2 & 3 & 1\\ 0  & -1 & 2 \end{bmatrix}\,\,\,\,\,$and 
$\,\,\,\,\,\textbf{B} = \begin{bmatrix} 2 & -2  \\ 4  & 3 \\1 & 5 \end{bmatrix}$. \,\,\,\,\,\,\textbf{A} is $2\times3$ and \textbf{B} is $3\times2$.     


    $\textbf{A}+\textbf{B}$ is undefined because the two matrices are incompatible for addition. 

 \[ \textbf{AB}=\begin{bmatrix} 2 & 3 & 1\\ 0  & -1 & 2 \end{bmatrix}\cdot\begin{bmatrix} 2 & -2  \\ 4  & 3 \\1 & 5 \end{bmatrix}=\begin{bmatrix} 2\cdot2+3\cdot4+1\cdot1 & 2\cdot (-2)+3\cdot3+1\cdot 5  \\0\cdot2+(-1)\cdot4 +2\cdot 1 & 0\cdot(-2)+(-1)\cdot3 +2\cdot5 \end{bmatrix}=\begin{bmatrix} 17 & 10  \\-2  & 7 \end{bmatrix} \]


 \[ \textbf{BA}=\begin{bmatrix} 2 & -2  \\ 4  & 3 \\1 & 5 \end{bmatrix}\cdot \begin{bmatrix} 2 & 3 & 1\\ 0  & -1 & 2 \end{bmatrix}=\begin{bmatrix} 2\cdot2+(-2)\cdot0 & 2\cdot 3+(-2)\cdot(1) &2\cdot1+(-2)\cdot 2  \\4\cdot2+3\cdot0&4\cdot3 +3\cdot (-1) & 4\cdot1+3\cdot2\\ 1\cdot2+5\cdot0&1\cdot3+5\cdot(-1)&1\cdot1+5\cdot2 \end{bmatrix}=\begin{bmatrix} 4 & 8&-2  \\8  & 9&10\\2&-2&11 \end{bmatrix} \]


\item %PROBLEM 2
  Let $\textbf{a}' = \begin{bmatrix}
       3 & 6 &-3 & 5 &  9& 2  \\
     \end{bmatrix}$. Find \textbf{a}$'$\textbf{a} and \textbf{aa}$'$.


 \textbf{a}$'$ is the transpose of $\textbf{a} = \begin{bmatrix}
       3 \\ 6 \\-3 \\ 5 \\  9\\ 2  \\
     \end{bmatrix}$

 \[\textbf{a}'\textbf{a} = \begin{bmatrix}
       3 & 6 &-3 & 5 &  9& 2  \\
     \end{bmatrix} \cdot \begin{bmatrix}
       3 \\ 6 \\-3 \\ 5 \\  9\\ 2  \\
     \end{bmatrix} = \begin{bmatrix}
       3^2 + 6^2 + (-3)^2 + 5^2 +  9^2 + 2^2  \\
     \end{bmatrix} =[164] \]

   \[\textbf{aa}' = \begin{bmatrix}
       3 \\ 6 \\-3 \\ 5 \\  9\\ 2  \\
     \end{bmatrix}\cdot \begin{bmatrix}
       3 & 6 &-3 & 5 &  9& 2  \\
     \end{bmatrix}  = \begin{bmatrix}
       3^2& 3\cdot 6 & 3\cdot (-3) & 3\cdot 5 &  3\cdot 9 & 3\cdot 2  \\
  6\cdot 3& 6^2 & 6\cdot (-3) & 6\cdot 5 &  6\cdot 9 & 6\cdot 2  \\
    (-3)\cdot 3& (-3)\cdot 6 & (-3)^2 & (-3)\cdot 5 &  (-3)\cdot 9 & (-3)\cdot 2  \\
      5\cdot 3& 5\cdot 6 & 5\cdot (-3) & 5^2 &  5\cdot 9 & 5\cdot 2  \\
        9\cdot 3& 9\cdot 6 & 9\cdot (-3) & 9\cdot 5 &  9^2 & 9\cdot 2  \\
         2\cdot 3& 2\cdot 6 & 2\cdot (-3) & 2\cdot 5 &  2\cdot 9 & 2^2  \\
     \end{bmatrix} \\=\begin{bmatrix}
       9& 18 & 9 & 15 &  27 & 6  \\
  18& 36 & -18 & 30 &  54 & 12  \\
    -9& -18 & 9 & -15 &  27 & -6  \\
      15& 30 & 15 & 10 &  45 & 10 \\
        27& 54 & 27 & 45 &  81 & 18  \\
         6& 12 & 6 & 10 &  18 & 4  \\
     \end{bmatrix} \]   







   Let $\textbf{b}' = \begin{bmatrix}
       1 & 1 & 1 & 1 & 1 & 1  \\
     \end{bmatrix}\,\,\,\,\,$
     and
    $\,\,\,\,\,\textbf{c}' = \begin{bmatrix}
       1 & -1 & 1 & -1 & 1 & -1  \\
     \end{bmatrix}$. Find \textbf{b}$'$\textbf{c} and \textbf{bc}$'$.


  \end{enumerate}
\end{document}   

Answer 1

I basically enacted two changes from your MWE. First, I used align=left in the enumerate call. Then, I replaced instances of \[ ... \] with $ ...$. Note that in some cases, you may have to invoke \displaystyle inside the $ delimiters, but that was not the case here.

Also, for math expressions that wrapped, I added few phantoms to get the proper indents.

\documentclass[paper=a4, fontsize=11pt]{scrartcl} % A4 paper an
\usepackage{amsmath,amsthm,amssymb}
\usepackage{enumitem}
\begin{document}

\begin{enumerate}[label=\bfseries Problem \arabic*:, align=left]
\parskip 1em
\item %PROBLEM 1

Find $\textbf{A}+\textbf{B}$, $\textbf{AB}$, $\textbf{BA}$.\\

\item[a)] $\textbf{A} = \begin{bmatrix}  1 & 2  \\   3  & -1  \end{bmatrix}\,\,\,\,\,$ and $\,\,\,\,\,\textbf{B} = \begin{bmatrix} 2 & 1  \\1  & 1 \end{bmatrix}$. \,\,\,\,\,\textbf{A} is $2\times2$ and \textbf{B} is $2\times2$.

 $\textbf{A}+\textbf{B}=\begin{bmatrix}  1 & 2  \\ 3  & -1\end{bmatrix} + \begin{bmatrix} 2 & 1  \\1  & 1 \end{bmatrix}=\begin{bmatrix} 1+2 & 2+1  \\3+1  & -1+1 \end{bmatrix}=\begin{bmatrix} 3 & 3  \\4  & 0 \end{bmatrix}$

 $\textbf{AB}=\begin{bmatrix}  1 & 2  \\ 3  & -1\end{bmatrix} \cdot \begin{bmatrix} 2 & 1  \\1  & 1 \end{bmatrix}=\begin{bmatrix} 1\cdot2+2\cdot1 & 1\cdot 1+2\cdot1  \\3\cdot2+(-1)\cdot1  & 3\cdot1+(-1)\cdot1 \end{bmatrix}=\begin{bmatrix} 4 & 3  \\5  & 2 \end{bmatrix}$

 $\textbf{BA}= \begin{bmatrix} 2 & 1  \\1  & 1 \end{bmatrix}\cdot \begin{bmatrix}  1 & 2  \\ 3  & -1\end{bmatrix}=\begin{bmatrix} 2\cdot1+1\cdot3 & 2\cdot 2+1\cdot(-1)  \\1\cdot1+1\cdot3  & 1\cdot2+1\cdot(-1) \end{bmatrix}=\begin{bmatrix} 5 & 3  \\4  & 1 \end{bmatrix}$


\item[b)] $\textbf{A} = \begin{bmatrix} 2 & 3 & 1\\ 0  & -1 & 2 \end{bmatrix}\,\,\,\,\,$and 
$\,\,\,\,\,\textbf{B} = \begin{bmatrix} 2 & -2  \\ 4  & 3 \\1 & 5 \end{bmatrix}$. \,\,\,\,\,\,\textbf{A} is $2\times3$ and \textbf{B} is $3\times2$.     


    $\textbf{A}+\textbf{B}$ is undefined because the two matrices are incompatible for addition. 

$\textbf{AB}=\begin{bmatrix} 2 & 3 & 1\\ 0  & -1 & 2 \end{bmatrix}\cdot\begin{bmatrix} 2 & -2  \\ 4  & 3 \\1 & 5 \end{bmatrix}=\begin{bmatrix} 2\cdot2+3\cdot4+1\cdot1 & 2\cdot (-2)+3\cdot3+1\cdot 5  \\0\cdot2+(-1)\cdot4 +2\cdot 1 & 0\cdot(-2)+(-1)\cdot3 +2\cdot5 \end{bmatrix}=\phantom{\textbf{AB}=}\begin{bmatrix} 17 & 10  \\-2  & 7 \end{bmatrix}$


 $\textbf{BA}=\begin{bmatrix} 2 & -2  \\ 4  & 3 \\1 & 5 \end{bmatrix}\cdot \begin{bmatrix} 2 & 3 & 1\\ 0  & -1 & 2 \end{bmatrix}=\begin{bmatrix} 2\cdot2+(-2)\cdot0 & 2\cdot 3+(-2)\cdot(1) &2\cdot1+(-2)\cdot 2  \\4\cdot2+3\cdot0&4\cdot3 +3\cdot (-1) & 4\cdot1+3\cdot2\\ 1\cdot2+5\cdot0&1\cdot3+5\cdot(-1)&1\cdot1+5\cdot2 \end{bmatrix}=\phantom{\textbf{BA}=}\begin{bmatrix} 4 & 8&-2  \\8  & 9&10\\2&-2&11 \end{bmatrix} $


\item %PROBLEM 2
  Let $\textbf{a}' = \begin{bmatrix}
       3 & 6 &-3 & 5 &  9& 2  \\
     \end{bmatrix}$. Find \textbf{a}$'$\textbf{a} and \textbf{aa}$'$.


 \textbf{a}$'$ is the transpose of $\textbf{a} = \begin{bmatrix}
       3 \\ 6 \\-3 \\ 5 \\  9\\ 2  \\
     \end{bmatrix}$

 \[\textbf{a}'\textbf{a} = \begin{bmatrix}
       3 & 6 &-3 & 5 &  9& 2  \\
     \end{bmatrix} \cdot \begin{bmatrix}
       3 \\ 6 \\-3 \\ 5 \\  9\\ 2  \\
     \end{bmatrix} = \begin{bmatrix}
       3^2 + 6^2 + (-3)^2 + 5^2 +  9^2 + 2^2  \\
     \end{bmatrix} =[164] \]

   $\textbf{aa}' = \begin{bmatrix}
       3 \\ 6 \\-3 \\ 5 \\  9\\ 2  \\
     \end{bmatrix}\cdot \begin{bmatrix}
       3 & 6 &-3 & 5 &  9& 2  \\
     \end{bmatrix}  = \begin{bmatrix}
       3^2& 3\cdot 6 & 3\cdot (-3) & 3\cdot 5 &  3\cdot 9 & 3\cdot 2  \\
  6\cdot 3& 6^2 & 6\cdot (-3) & 6\cdot 5 &  6\cdot 9 & 6\cdot 2  \\
    (-3)\cdot 3& (-3)\cdot 6 & (-3)^2 & (-3)\cdot 5 &  (-3)\cdot 9 & (-3)\cdot 2  \\
      5\cdot 3& 5\cdot 6 & 5\cdot (-3) & 5^2 &  5\cdot 9 & 5\cdot 2  \\
        9\cdot 3& 9\cdot 6 & 9\cdot (-3) & 9\cdot 5 &  9^2 & 9\cdot 2  \\
         2\cdot 3& 2\cdot 6 & 2\cdot (-3) & 2\cdot 5 &  2\cdot 9 & 2^2  \\
     \end{bmatrix} \\\phantom{\textbf{aa}'}=\begin{bmatrix}
       9& 18 & 9 & 15 &  27 & 6  \\
  18& 36 & -18 & 30 &  54 & 12  \\
    -9& -18 & 9 & -15 &  27 & -6  \\
      15& 30 & 15 & 10 &  45 & 10 \\
        27& 54 & 27 & 45 &  81 & 18  \\
         6& 12 & 6 & 10 &  18 & 4  \\
     \end{bmatrix} $







   Let $\textbf{b}' = \begin{bmatrix}
       1 & 1 & 1 & 1 & 1 & 1  \\
     \end{bmatrix}\,\,\,\,\,$
     and
    $\,\,\,\,\,\textbf{c}' = \begin{bmatrix}
       1 & -1 & 1 & -1 & 1 & -1  \\
     \end{bmatrix}$. Find \textbf{b}$'$\textbf{c} and \textbf{bc}$'$.


  \end{enumerate}
\end{document}   

Answer 2

Here is a solution: since you load amsthm I define problem as a example-style new theorem. This cater fort automatic numbering and cross referencing of problems. Also, the questions are typeset in italic. If you prefer roman shape, change example style for definition style.

Also, as the vertical spacing between theorems seems too tight, I added some vrtical spacing, with the help of the etoolbox package, more precisely its \AtBeginEnvironment command.

As for the alignments, I use the align* environment. Some matrices look better (those with some negative coefficients), in my opinion, if their columns are right-aligned. For these, I used the bmatrix* environment, from mathtools, with option [r]. Don't load amsmath: mathtools does it for you.

\documentclass[paper=a4, fontsize=11pt]{scrartcl} % A4 paper

\usepackage{mathtools, amsthm, amssymb}
\usepackage{enumitem}
\usepackage{etoolbox}

\theoremstyle{example}
\newtheorem{problem}{Problem}
\AtBeginEnvironment{problem}{\vspace{1ex}}{}{}

\begin{document}

\begin{problem}\label{pb1}% PROBLEM 1
Find $\mathbf{A}+\mathbf{B}$, $\mathbf{AB}$, $\mathbf{BA}$.
\end{problem}

 \begin{enumerate}[label = \alph*), wide = 0em]
\item $\mathbf{A} = \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix}$\enspace and \enspace$\mathbf{B} = \begin{bmatrix} 2 & 1 \\1 & 1 \end{bmatrix}$. $ \mathbf{A} $ is $2\times2$ and $ \mathbf{B} $ is $2\times2$.

\begin{align*}
    \mathbf{A}+\mathbf{B} &= \begin{bmatrix*}[r] 1 & 2 \\ 3 & -1\end{bmatrix*} + \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix*}[r] 1+2 & 2+1 \\ 3+1 & -1+1 \end{bmatrix*}= \begin{bmatrix} 3 & 3 \\ 4 & 0 \end{bmatrix}\\[1ex]
%
  \mathbf{AB} &= \begin{bmatrix*}[r] 1 & 2 \\ 3 & -1\end{bmatrix*} \cdot \begin{bmatrix} 2 & 1 \\1 & 1 \end{bmatrix}= \begin{bmatrix} 1\cdot2+2\cdot1 & 1\cdot 1+2\cdot1 \\3\cdot2+(-1)\cdot1 & 3\cdot1+(-1)\cdot1 \end{bmatrix} = \begin{bmatrix} 4 & 3 \\ 5 & 2 \end{bmatrix} \\[1ex]
%
  \mathbf{BA} &= \begin{bmatrix} 2 & 1 \\1 & 1 \end{bmatrix}\cdot \begin{bmatrix*}[r] 1 & 2 \\ 3 & -1\end{bmatrix*} = \begin{bmatrix} 2\cdot1+1\cdot3 & 2\cdot 2+1\cdot(-1) \\1\cdot1+1\cdot3 & 1\cdot2+1\cdot(-1) \end{bmatrix} = \begin{bmatrix} 5 & 3 \\ 4 & 1 \end{bmatrix} \\
\end{align*}

\item $\mathbf{A} = \begin{bmatrix} 2 & 3 & 1\\ 0 & -1 & 2 \end{bmatrix}$\enspace and\enspace$ \mathbf{B} = \begin{bmatrix} 2 & -2 \\ 4 & 3 \\1 & 5 \end{bmatrix}$. $ \mathbf{A} $ is $2\times3$ and $ \mathbf{B} $ is $3\times2$.

    $\mathbf{A}+\mathbf{B}$ is undefined because the two matrices are incompatible for addition.

\begin{align*}
   \mathbf{AB} = \begin{bmatrix*}[r] 2 & 3 & 1 \\ 0 & -1 & 2 \end{bmatrix*} \cdot
   \smash{\begin{bmatrix*}[r] 2 & -2 \\ 4 & 3 \\ 1 & 5 \end{bmatrix*}}
   & = \begin{bmatrix}
   2\cdot 2+3\cdot 4+1\cdot 1 & 2\cdot (-2)+3\cdot 3+1\cdot 5 \\
   0\cdot2+(-1)\cdot 4 +2\cdot 1 & 0\cdot (-2)+(-1)\cdot 3 +2\cdot 5
   \end{bmatrix}\\
    & = \begin{bmatrix*}[r] 17 & 10 \\-2 & 7 \end{bmatrix*}\\[1.5ex]
%
   \mathbf{BA} = \begin{bmatrix*}[r] 2 & -2 \\ 4 & 3 \\1 & 5 \end{bmatrix*}\cdot
   \begin{bmatrix*}[r] 2 & 3 & 1\\ 0 & -1 & 2 \end{bmatrix*}
   & = \begin{bmatrix}
   2\cdot 2+(-2)\cdot 0 & 2\cdot 3+(-2)\cdot (-1) & 2\cdot 1+(-2)\cdot 2 \\4\cdot 2+3\cdot 0&4\cdot3 +3\cdot (-1) & 4\cdot1+3\cdot2 \\
   1\cdot 2+5\cdot 0& 1\cdot3 + 5\cdot (-1)&1\cdot 1+5\cdot 2
   \end{bmatrix}\\
    & = \begin{bmatrix*}[r] 4 & 8 & -2 \\
    8 & 9 & 10\\ 2 & -2 & 11
    \end{bmatrix*}
\end{align*}

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem}\label{pb2} %PROBLEM 2
  Let $\mathbf{a}' = \begin{bmatrix}3 & 6 &-3 & 5 & 9& 2 \\ \end{bmatrix}$. Find $\mathbf{a}'\mathbf{a}$ and $\mathbf{aa}'$.
\end{problem}

$\mathbf{a}'$ is the transpose of $\mathbf{a} = \begin{bmatrix}3 \\ 6 \\-3 \\ 5 \\ 9\\ 2 \end{bmatrix}$.

\begin{align*}
  \mathbf{a}' \mathbf{a} & = \begin{bmatrix} 3 & 6 &-3 & 5 & 9& 2 \end{bmatrix} \cdot \begin{bmatrix}
       3 \\ 6 \\-3 \\ 5 \\ 9\\ 2 \end{bmatrix} = \begin{bmatrix} 3^2 + 6^2 + (-3)^2 + 5^2 + 9^2 + 2^2
     \end{bmatrix} = [164] \\
%
  \mathbf{aa}' & = \begin{bmatrix} 3 \\ 6 \\-3 \\ 5 \\ 9\\ 2 \end{bmatrix}\cdot \begin{bmatrix} 3 & 6 &-3 & 5 & 9& 2 \end{bmatrix}
   = \begin{bmatrix}
       3^2& 3\cdot 6 & 3\cdot (-3) & 3\cdot 5 & 3\cdot 9 & 3\cdot 2 \\
  6\cdot 3& 6^2 & 6\cdot (-3) & 6\cdot 5 & 6\cdot 9 & 6\cdot 2 \\
    (-3)\cdot 3& (-3)\cdot 6 & (-3)^2 & (-3)\cdot 5 &( -3)\cdot 9 & (-3)\cdot 2 \\
      5\cdot 3& 5\cdot 6 & 5\cdot (-3) & 5^2 & 5\cdot 9 & 5\cdot 2 \\
        9\cdot 3& 9\cdot 6 & 9\cdot (-3) & 9\cdot 5 & 9^2 & 9\cdot 2 \\
         2\cdot 3& 2\cdot 6 & 2\cdot (-3) & 2\cdot 5 & 2\cdot 9 & 2^2
     \end{bmatrix} \\
      & = \begin{bmatrix*}[r]
       9& 18 & 9 & 15 & 27 & 6 \\
  18& 36 & -18 & 30 & 54 & 12 \\
    -9& -18 & 9 & -15 & 27 & -6 \\
      15& 30 & 15 & 10 & 45 & 10 \\
        27& 54 & 27 & 45 & 81 & 18 \\
         6& 12 & 6 & 10 & 18 & 4
     \end{bmatrix*}
\end{align*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem}\label{pb3}%% PROBLEM 3
       Let $\mathbf{b}' = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 \end{bmatrix}$\enspace and\enspace$ \mathbf{c}' = \begin{bmatrix} 1 & -1 & 1 & -1 & 1 & -1 \end{bmatrix}$. Find $\mathbf{b}'\mathbf{c}$ and $\mathbf{bc}'$.
\end{problem}

\end{document} 

First page of the resulting .pdf: