= Solution
Assume the composite <pure state> has unit norm. Put its amplitudes into the <matrix>
$$
C=\begin{pmatrix}a_1&a_2\\a_3&a_4\end{pmatrix},
\qquad |\chi\rangle=\sum_{s,t}C_{st}|s\rangle_A|t\rangle_B.
$$
The <partial trace> contracts the same $B$ index between ket and bra:
$$
(\rho_A)_{ss'}=\sum_t C_{st}C^*_{s't}.
$$
Thus
$$
\boxed{\rho_A=CC^\dagger=
\begin{pmatrix}
|a_1|^2+|a_2|^2&a_1a_3^*+a_2a_4^*\\
a_1^*a_3+a_2^*a_4&|a_3|^2+|a_4|^2
\end{pmatrix}.}
$$
It is <Hermitian>, <positive semidefinite> and has <trace> one. Its <determinant> is particularly useful:
$$
\boxed{\det\rho_A=|\det C|^2=|a_1a_4-a_2a_3|^2.}
$$
This follows from multiplicativity of the <determinant>, or by expanding the two-by-two expression. If the supplied amplitudes were not normalized, the physical reduced state would be $CC^\dagger/\operatorname{tr}(CC^\dagger)$ instead.
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