I am trying to understand the relation between algorithmic complexity and circuit complexity of Determinants and Matrix Multiplication.It is known that the determinant of an $n\times n$ matrix can be computed in $\tilde{O}(M(n))$ time, where $M(n)$ is the minimum time required to multiply any two $n\times n$ matrices.  It is also known that the best circuit complexity of determinants is polynomial at depth $O(\log^{2}(n))$ and exponential at depth 3.  But the circuit complexity of matrix multiplication, for any constant depth, is only polynomial.Why is there a difference in circuit complexity for determinants and matrix multiplication while it is known that from an algorithm perspective determinant calculation is similar to matrix multiplication? Specifically, why do the circuit complexities have an exponential gap at depth-$3$? Probably, the explanation is simple but I do not see it. Is there an explanation with 'rigor'?Also look in: Smallest known formula for the determinant
Determinants and Matrix Multiplication - Similarity and differences in algorithmic complexity and arithmetic circuit size
algebraic complexity;arithmetic circuits;matrix product;determinant