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We give an efficient solution to the following problem: Given $X_1, \ldotsX_d$ and $Y$ some $n$ by $n$ matrices can we determine if $Y$ is in the unitalalgebra generated by $X_1, \ldots, X_d$ as a subalgebra of all $n$ by $n$matrices? The solution also gives an easy method for computing the dimension ofthis algebra.

An elementary method to compute the algebra generated by some given matrices and its dimension