there is a well-defined notion of equality among matrices, as there is among vectors in R^n: to wit, two matrices are equal if and only if every i,j-th entry is equal for both matrices.

as matrices often occur in equations, where the entries of a matrix, or of a vector a matrix acts upon is unknown, "identically equal to" isn't usually appropriate, unless one is talking about a certain specially defined matrix, such as the nxn identity matrix.

furthermore, there are different notions of "equivalence" of matrices, the most important of which is probably similarity.