1. Matrix addition and subtraction: Two identical matrices can be added or subtracted, and the result is still the same matrix. The arithmetic of matrix addition and subtraction is the addition or subtraction of corresponding elements.

2. Matrix multiplication: Two matrices can be multiplied and the result is a high-order matrix. The operation rule of matrix multiplication is dot product operation according to rows and columns.

3. Matrix transpose: A matrix can be transposed, and the result is the transposed matrix of the original matrix. The operation rule of matrix transposition is to exchange the rows and columns of the original matrix.

4. Matrix Inversion: If a square matrix satisfies the condition that the determinant is not 0, then it can be inversed, and the result is the inverse of the original matrix. The operation rules of matrix inversion are solved by Gauss-Jordan elimination method or adjoint matrix method.

5. Rank of a matrix: the rank of a matrix refers to the dimension of its row space or column space. The rank of matrix can be solved by row transformation or column transformation.

6. Eigenvalues and eigenvectors of matrices: For a square matrix, if there is a non-zero vector V and a real number λ, so that Av=λv, then we call λ the eigenvalue of A and V the corresponding eigenvector.

7. Singular Value Decomposition (SVD) of Matrix: For any m×n matrix A, we can decompose it into the product of three matrices A = u σ v t, where u and v are orthogonal matrices and σ is diagonal matrix.

The above are common matrix operations in advanced mathematics, which are widely used in solving practical problems.