This paper analyzes the hardness of the Matrix Code Equivalence (MCE) problem for matrix codes under the rank metric and presents the first algorithms for solving it. The MCE problem is shown to be polynomial-time equivalent to the Bilinear Maps Linear Equivalence (BMLE) and Quadratic Maps Linear Equivalence (QMLE) problems under the assumption of trivial automorphism groups. These reductions establish that MCE is at least as hard as QMLE, which is considered the hardest equivalence problem for systems of multivariate polynomials. The paper also shows that MCE is related to other code equivalence problems, including the Vector Sum-Rank Code Equivalence (VSRCE) and Matrix Sum-Rank Code Equivalence (MSRCE) problems. On the practical side, two algorithms are presented: a probabilistic algorithm with time complexity $ q^{\frac{2}{3}(n+m)} $ and a deterministic algorithm with time complexity $ q^{\min\{m,n,k\}} $, both up to polynomial factors. These algorithms are tested on randomly generated instances of MCE, confirming their effectiveness. The results highlight the importance of understanding the hardness of MCE for cryptographic applications, particularly in post-quantum cryptography. The paper also discusses the connection between MCE and the Isomorphism of Polynomials (IP) problem, showing that MCE can be reduced to IP1S, which is known to be polynomially solvable for many instances. The findings contribute to the understanding of the security of cryptographic schemes based on code equivalence problems.This paper analyzes the hardness of the Matrix Code Equivalence (MCE) problem for matrix codes under the rank metric and presents the first algorithms for solving it. The MCE problem is shown to be polynomial-time equivalent to the Bilinear Maps Linear Equivalence (BMLE) and Quadratic Maps Linear Equivalence (QMLE) problems under the assumption of trivial automorphism groups. These reductions establish that MCE is at least as hard as QMLE, which is considered the hardest equivalence problem for systems of multivariate polynomials. The paper also shows that MCE is related to other code equivalence problems, including the Vector Sum-Rank Code Equivalence (VSRCE) and Matrix Sum-Rank Code Equivalence (MSRCE) problems. On the practical side, two algorithms are presented: a probabilistic algorithm with time complexity $ q^{\frac{2}{3}(n+m)} $ and a deterministic algorithm with time complexity $ q^{\min\{m,n,k\}} $, both up to polynomial factors. These algorithms are tested on randomly generated instances of MCE, confirming their effectiveness. The results highlight the importance of understanding the hardness of MCE for cryptographic applications, particularly in post-quantum cryptography. The paper also discusses the connection between MCE and the Isomorphism of Polynomials (IP) problem, showing that MCE can be reduced to IP1S, which is known to be polynomially solvable for many instances. The findings contribute to the understanding of the security of cryptographic schemes based on code equivalence problems.