2. The eigenvalues of identity matrix are all 1, and any vector is the eigenvector of identity matrix.
3. Because the product of eigenvalues is equal to determinant, the determinant of identity matrix is 1. Because the sum of eigenvalues is equal to the number of traces, the trace of identity matrix is n.
4. When two rows are exchanged, the determinant changes sign.
5. One row of the matrix subtracts the multiple of another row, and the determinant remains unchanged.
6. If the matrix is triangular, then the determinant is equal to the product of the elements on the diagonal.
Extended data
Matrix is often used in applied mathematics such as statistical analysis. In physics, matrices have applications in circuit science, mechanics, optics and quantum physics. In computer science, three-dimensional animation also needs matrix. Matrix operation is an important problem in the field of numerical analysis. Decomposition of a matrix into a combination of simple matrices can simplify the operation of the matrix in theory and practical application.
For some widely used and special matrices, such as sparse matrix and quasi-diagonal matrix, there are concrete fast operation algorithms. For the development and application of matrix related theory, please refer to matrix theory. Infinite-dimensional matrices will also appear in astrophysics, quantum mechanics and other fields, which is the generalization of matrices.