  The component-wise or Schur product $C*C'$ of two linear error correcting
codes $C$ and $C'$ over certain finite field is the linear code spanned by all
component-wise products of a codeword in $C$ with a codeword in $C'$. When
$C=C'$, we call the product the square of $C$ and denote it $C^{*2}$. Motivated
by several applications of squares of linear codes in the area of cryptography,
in this paper we study squares of so-called matrix-product codes, a general
construction that allows to obtain new longer codes from several "constituent"
codes. We show that in many cases we can relate the square of a matrix-product
code to the squares and products of their constituent codes, which allow us to
give bounds or even determine its minimum distance. We consider the well-known
$(u,u+v)$-construction, or Plotkin sum (which is a special case of a
matrix-product code) and determine which parameters we can obtain when the
constituent codes are certain cyclic codes. In addition, we use the same
techniques to study the squares of other matrix-product codes, for example when
the defining matrix is Vandermonde (where the minimum distance is in a certain
maximal with respect to matrix-product codes).
