Final answer:
Matrix operations A + B and A - B are defined for matrices of the same dimensions. A * B is defined if the number of columns in A matches the number of rows in B. 'A / B' is not formally defined, but can be understood as A * B^-1 when B is invertible.
Step-by-step explanation:
Matrix operations such as addition, subtraction, and multiplication are defined under certain conditions, while division is not directly defined. The operations A + B and A - B are defined only if matrices A and B are of the same dimensions. The matrix subtraction can be viewed as adding A to the negative of B: A - B = A + (-B).
Matrix multiplication A * B is defined if the number of columns in A is equal to the number of rows in B. However, matrix division does not exist in the usual sense. Instead of dividing by a matrix, the concept of finding an inverse is used, provided the matrix is square and non-singular; then, for any matrix A, the expression A/B can also be represented by A * B-1, with B-1 being the inverse of B.
Scalar product (dot product) and cross product are operations specific to vectors. The scalar product is the product of the magnitudes of the two vectors and the cosine of the angle between them, and this operation is only for vectors, not matrices.