True Bit Complexity of matrix multiplication is $O(n^{4})$
No, the bit complexity of matrix multiplication on $M$-bit entries is $n^{\omega} (\log n)^{O(1)} \cdot M (\log M)^{O(1)}$, where $\omega &lt; 2.4$ is the best known matrix multiplication exponent. Multiplication and addition of $M$-bit numbers can be done in $M (\log M)^2$ time. Multiplying two $M$-bit numbers yields a number which has no more than $2M$ bits. Adding $n$ numbers of $M$ bits each, yields a number which has no more than $M+\log n+O(1)$ bits. (Think about it: the sum is at most $n 2^M$, so the bit representation takes no more than $\log (n 2^M)+O(1)$ bits.)

References to fast integer multiplication algorithms can be found with a web search or wikipedia.