= Solution
The <Artin–Wedderburn theorem> says that every <central simple algebra> $A$ over $k$ has the form $M_r(D)$ for a finite-dimensional central division algebra $D$, uniquely up to the evident data. If $A$ and $B$ are central simple, extend scalars to an algebraic closure $\bar k$. Both become full matrix algebras, hence
$$
(A\otimes_kB)\otimes_k\bar k
\cong M_{ab}(\bar k)
$$
for suitable $a,b$. Any nonzero proper ideal of $A\otimes_kB$ would extend to one in this simple matrix algebra, and faithful flatness prevents it from vanishing or becoming the whole algebra. The same scalar-extension argument shows that the center is $k$. This proves the <tensor product of central simple algebras> theorem.
The <Brauer group> $\operatorname{Br}(k)$ consists of <Morita equivalence> classes of central simple $k$-algebras. Its product is $[A][B]=[A\otimes_kB]$, its identity is $[k]$, and $[A]^{-1}=[A^{\mathrm{op}}]$ because $A\otimes_kA^{\mathrm{op}}\cong\operatorname{End}_k(A)$ is a full matrix algebra.
Let $L/k$ be a <Finite Galois extension> with <Galois group> $G_L$, and let $\phi:G_L\times G_L\to L^\times$ be a <normalized two-cocycle>. The <crossed-product algebra of a Galois extension> has underlying left $L$-vector space
$$
A(L,G_L,\phi)=\bigoplus_{\sigma\in G_L}Lu_\sigma
$$
and multiplication
$$
u_\sigma a=\sigma(a)u_\sigma,
\qquad
u_\sigma u_\tau=\phi(\sigma,\tau)u_{\sigma\tau}.
$$
The cocycle identity is exactly associativity. After scalar extension to $L$, the algebra acts by the twisted regular representation and becomes $M_{|G_L|}(L)$; Galois descent shows that it is central simple over $k$. If $\phi$ is multiplied by the coboundary of a one-cochain $b_\sigma$, rescaling $u_\sigma$ by $b_\sigma$ gives an isomorphic algebra. Hence the <cohomological construction of a Brauer class> gives a well-defined map
$$
H^2(G_L,L^\times)\longrightarrow\operatorname{Br}(k).
$$
It remains to show that every Brauer class is torsion. For a finite group $G$, restriction and corestriction on normalized bar cochains satisfy
$$
\operatorname{cor}\circ\operatorname{res}
=\sum_{g\in G}g^*=|G|
$$
on cohomology: the first equality follows by summing the translated cochain over coset representatives, and each $g^*$ is the identity because an inner automorphism is cochain-homotopic to the identity. Restriction to the trivial subgroup is zero in positive degree, so this proves that <finite-group cohomology is annihilated by the group order>. In multiplicative notation, every $\alpha\in H^2(G_L,L^\times)$ therefore satisfies $\alpha^{|G_L|}=1$.
By the permitted assumption, $[A]$ is the image of such an $\alpha$ for some $L$. Consequently $[A]^{|G_L|}=1$ in $\operatorname{Br}(k)$. By the definition of Brauer equivalence, this says that for some $n$,
$$
A^{\otimes|G_L|}\cong M_n(k),
$$
as required.
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