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Given an unknown input vector $V= (v_1, v_2, v_3, v_4)$, a known $4\times 4$ matrix $A$ and a known vector-matrix product $M=[m_1,...,m_4]$. Can you discover $V$?

Normally you would just take the inverse of $A$, and right multiply it with $M$ to get $V$. However, here's the trick - in this environment, you are not allowed to right multiply, only left multiply with $4 \times 4$ matrices.

Is there a sequence of left multiplication operations that will produce the original vector? (I don't think so, but I had to ask)

(Note - no transpose operations are permitted: only $4\times 4$ multiplication)

M47145
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Michael Mullany
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1 Answers1

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If you can compute $A^{-1}$, you can also compute transpose. So, compute $$V=MA^{-1}=\left( (A^{-1})^T\,M^T \right)^T\,.$$ Here, if I understand correctly, $M$ is a row vector, so $M^T$ will be the corresponding column vector.

Berci
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