In wiki, it is shown that the time complexity of matrix multiplication and matrix inverse is similar. But people always to argue it is easier to do matrix multiplication rather than inverse. Is this claim true? Maybe we need to consider various computation platform, CPU or GPU.
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Matrix multiplication and matrix inverse have the same asymptotic running time. If we denote the running time of multiplying two $n \times n$ matrices by $T_1(n)$, and that of inverting an $n \times n$ matrix by $T_2(n)$, then this means that there exist constants $A,B$ such that $$ T_1(n) \leq AT_2(n) \\ T_2(n) \leq BT_1(n). $$ However, the constants $A,B$ can be significant. (Also, numerical stability might be an issue.)
Checking your claim is, however, very easy in practice. Choose your favorite matrix library (e.g. LAPACK), and you can determine empirically $T_1(n),T_2(n)$ for the values of $n$ that you are interested in, on the hardware platform that you are interested in.
answered May 2, 2020 at 9:33