Matrix contract can draw the conclusion that the positive and negative inertia indexes are the same, and the positive inertia index and rank are the same. Different from similarity conclusion, similarity is related to eigenvalue. The eigenvalues are the same, the determinant is the same, similarity is a contract, and the sum of the main diagonals of the two matrices is the same. If the matrices are similar, they represent the same linear transformation in different coordinate systems (bases).

Non-homogeneous linear equations, AX=B, when a and its augmented matrix rank = r

Application of matrix

After the study of determinant, matrix officially appeared as the research object of mathematics. Logically, the concept of matrix precedes determinant, but in actual history it is just the opposite. Japanese mathematician Guan Xiaohe (1683) and Gottfried Wilhelm Leibniz (1693), one of the discoverers of calculus, established determinant theory almost simultaneously.

Then the determinant gradually developed into a tool for solving linear equations. 1750, Gabriel Cramer discovered Cramer's law. After the study of determinant, matrix officially appeared as the research object of mathematics. Logically, the concept of matrix precedes determinant, but in actual history it is just the opposite.

Japanese mathematician Guan Xiaohe (1683) and Gottfried Wilhelm Leibniz (1693), one of the discoverers of calculus, established determinant theory almost simultaneously. Then the determinant gradually developed into a tool for solving linear equations. 1750, Gabriel Cramer discovered Cramer's law.