This paper analyzes the hardness of the Matrix Code Equivalence (MCE) problem for matrix codes endowed with the rank metric and provides the first algorithms for solving it. The authors establish connections between MCE and the Isomorphism of Polynomials (IP) problem, showing tight reductions from MCE to the homogeneous version of the Quadratic Maps Linear Equivalence (QMLE) problem and vice versa. They also present reductions to and from similar problems in the sum-rank metric, demonstrating that MCE is central to code equivalence problems. Practically, they introduce two algorithms: a probabilistic algorithm for MCE with a complexity of \( q^{\frac{2}{3}(n+m)} \) up to a polynomial factor, and a deterministic algorithm for MCE with roots, running in time \( q^{\min(m,n,k)} \) up to a polynomial factor. These algorithms are validated through experiments on randomly generated instances of MCE. The paper also discusses the practical hardness of MCE and its variants, such as MCEbase and MCRE, and provides reductions to and from other code equivalence problems, including Hamming Code Equivalence (HCE) and Sum-Rank Code Equivalence (SRCE).This paper analyzes the hardness of the Matrix Code Equivalence (MCE) problem for matrix codes endowed with the rank metric and provides the first algorithms for solving it. The authors establish connections between MCE and the Isomorphism of Polynomials (IP) problem, showing tight reductions from MCE to the homogeneous version of the Quadratic Maps Linear Equivalence (QMLE) problem and vice versa. They also present reductions to and from similar problems in the sum-rank metric, demonstrating that MCE is central to code equivalence problems. Practically, they introduce two algorithms: a probabilistic algorithm for MCE with a complexity of \( q^{\frac{2}{3}(n+m)} \) up to a polynomial factor, and a deterministic algorithm for MCE with roots, running in time \( q^{\min(m,n,k)} \) up to a polynomial factor. These algorithms are validated through experiments on randomly generated instances of MCE. The paper also discusses the practical hardness of MCE and its variants, such as MCEbase and MCRE, and provides reductions to and from other code equivalence problems, including Hamming Code Equivalence (HCE) and Sum-Rank Code Equivalence (SRCE).