The inverse matrix is a very important concept in linear algebra. It has the following basic properties:
1. An invertible matrix must be a square matrix: If a matrix is to be invertible, it must be a square matrix, that is, its number of rows and columns are equal.
2. The uniqueness of the inverse matrix: If the matrix \( A \) is invertible, then its inverse matrix \( A^{-1} \) is unique. This means that in the same number field, there are no two different inverse matrices multiplied by \( A \) that will result in the identity matrix \( I \).
3. The inverse of an inverse matrix is still the original matrix: for any invertible matrix \( A \), the inverse matrix of its inverse matrix \( A^{-1} \) is still \( A \), that is, \( (A ^{-1})^{-1} = A \).
4. An invertible matrix is still invertible after transposition: If the matrix \( A \) is invertible, then its transposed matrix \( A^T \) is also invertible, and \( (A^T)^{-1} = (A^{-1})^T \).
5. Elimination law: If the matrix \( A \) is invertible, then for any matrix \( B \) and \( C \), if \( AB = C \), then \( B = C \cdot A^ {-1} \). Similarly, if \( BA = C \), then \( A = C \cdot B^{-1} \).
6. The product of two invertible matrices is still invertible: If the matrices \( A \) and \( B \) are both invertible, then their product \( AB \) is also invertible.
7. The determinant of an invertible matrix is not zero: The necessary and sufficient condition for a matrix to be invertible is that its determinant is not zero. If the determinant of matrix \( A \) \( \det(A) \neq 0 \), then \( A \) is invertible.
8. The rank of an invertible matrix is equal to its order: The rank of an invertible matrix is equal to its number of rows (or columns), which means that all its rows (or columns) are linearly independent.
These properties have very important applications in solving problems such as linear equations, calculating the inverse of a matrix, and performing matrix factorization. They are an indispensable tool in linear algebra and are crucial to understanding matrix theory and its applications.
