Itemize using multicol package

Here's a solution that doesn't use a multicols environment. Instead, it loads the enumitem package with the option inline and uses an enumerate* environment. The solution also employs the machinery of the amsthm package to streamline the definition of the problem environment.

\documentclass[12pt]{article}
%% (I've simplified the preamble code to the bare minimum)
\usepackage{amsmath,amssymb,amsthm}
\theoremstyle{definition} % upright-lettering style
\newtheorem{problem}{Problem}

\usepackage[inline]{enumitem} % <-- note the option 'inline'

\begin{document}

\setcounter{problem}{6} % adjust as needed
\begin{problem}
Calculate the following determinants.

\begin{enumerate*}[label=(\alph*)]
\item $\begin{vmatrix} 3 & 0 \\ 2 & 6 \end{vmatrix}$ 
\hspace*{1cm} %  choose a suitable amount of horizontal whitespace
\item $\begin{vmatrix} a & a \\ b & b \end{vmatrix}$ 
\hspace*{1cm}
\item $\begin{vmatrix} a+b & a-b \\ a-b & a+b \end{vmatrix}$ 
\hspace*{1cm}
\item $\begin{vmatrix} 3^t & 2^t \\ 3^{t-1} & 2^{t-1} \end{vmatrix}$
\end{enumerate*}
\end{problem}

\end{document}

Another fast solution could be:

1. to use \usepackage[margin=1in]{geometry};
2. enumitem inline \usepackage[inline]{enumitem};
3. To write correctly the matrices \begin{vmatrix}....\end{vmatrix};

---

\documentclass[12pt]{article}
\usepackage[margin=1in]{geometry}
\newenvironment{problem}[2][Problem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{multicol} % to insert columns
    \setlength{\columnsep}{5pt}
\usepackage{hyperref}
    \hypersetup{
        colorlinks=true,
        linkcolor=cyan,
        filecolor=magenta,      
        urlcolor=blue,
        }
\usepackage{parskip} % For no indentation and a bit of space in paragraphs

% to format enumerate to letters in line
\usepackage[inline]{enumitem}
\begin{document}

\title{Problem Set 2}
\author{Ruben Perez Sanz}
\date{7 September, 2020}
\maketitle


\begin{problem}{7}
    Calculate the following determinants:
    \begin{multicols}{4}
        \begin{enumerate}[label=(\alph*)]
            \item $\begin{vmatrix} 3 & 0 \\  2 & 6 \end{vmatrix}$
            \item $\begin{vmatrix} a & a \\  b & b \end{vmatrix}$
            \item $\begin{vmatrix} a+b & a-b \\  a-b & a+b \end{vmatrix}$
            \item $\begin{vmatrix} 3^t & 2^t \\  3^{t-1} & 2^{t-1} \end{vmatrix}$
        \end{enumerate}
    \end{multicols}
\end{problem}


\end{document}