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We consider the following variant of the Mortality Problem: given $k\times k$matrices $A_1, A_2, \dots,A_{t}$, does there exist nonnegative integers $m_1,m_2, \dots,m_t$ such that the product $A_1^{m_1} A_2^{m_2} \cdotsA_{t}^{m_{t}}$ is equal to the zero matrix? It is known that this problem isdecidable when $t \leq 2$ for matrices over algebraic numbers but becomesundecidable for sufficiently large $t$ and $k$ even for integral matrices.  In this paper, we prove the first decidability results for $t>2$. We show asone of our central results that for $t=3$ this problem in any dimension isTuring equivalent to the well-known Skolem problem for linear recurrencesequences. Our proof relies on the Primary Decomposition Theorem for matricesthat was not used to show decidability results in matrix semigroups before. Asa corollary we obtain that the above problem is decidable for $t=3$ and $k \leq3$ for matrices over algebraic numbers and for $t=3$ and $k=4$ for matricesover real algebraic numbers. Another consequence is that the set of triples$(m_1,m_2,m_3)$ for which the equation $A_1^{m_1} A_2^{m_2} A_3^{m_3}$ equalsthe zero matrix is equal to a finite union of direct products of semilinearsets.  For $t=4$ we show that the solution set can be non-semilinear, and thus itseems unlikely that there is a direct connection to the Skolem problem. Howeverwe prove that the problem is still decidable for upper-triangular $2 \times 2$rational matrices by employing powerful tools from transcendence theory such asBaker's theorem and S-unit equations.

On the Mortality Problem: from multiplicative matrix equations to linear recurrence sequences and beyond