\[\begin{pmatrix}
\dfrac{dx}{dt}\\[8pt]
\dfrac{dy}{dt}\\[8pt]
\dfrac{dz}{dt}\\[8pt]
\end{pmatrix}
=
\begin{pmatrix}
t & -t^2 & 1\\[8pt]
0 & \cos(t) & 1\\[8pt]
1 & 0 & 3\\[8pt]
\end{pmatrix}
\begin{pmatrix}
x\\[8pt]
y\\[8pt]
z\\[8pt]
\end{pmatrix}
+
\begin{pmatrix}
0\\[8pt]
0\\[8pt]
\tan(t)\\[8pt]
\end{pmatrix}
,\quad
\begin{pmatrix}
x(0)\\[8pt]
y(0)\\[8pt]
z(0)\\[8pt]
\end{pmatrix}
=
\begin{pmatrix}
1\\[8pt]
-1\\[8pt]
3\\[8pt]
\end{pmatrix}
\]
or it can be rewritten in the form \(\bf{X}'=\bf{AX}+\bf{b}\) as
\[
\bf{X}'
=
\begin{pmatrix}
t & -t^2 & 1\\[8pt]
0 & \cos(t) & 1\\[8pt]
1 & 0 & 3\\[8pt]
\end{pmatrix}
\bf{X}
+
\begin{pmatrix}
0\\[8pt]
0\\[8pt]
\tan(t)\\[8pt]
\end{pmatrix}
,\quad
\bf{X}(0)
=
\begin{pmatrix}
1\\[8pt]
-1\\[8pt]
3\\[8pt]
\end{pmatrix}
\]