Linear Algebra If A is a symmetric matrix, is the inverse matrix of A a symmetric matrix? Why?

If A is a symmetric matrix, the inverse matrix of A is also a symmetric matrix for the following reasons:

If a is a symmetric matrix, the transposed matrices of a and a are equal.

For the transposed matrix of A, its inverse matrix is equal to that of A, that is, the transposed matrix of A is equal to that of A. According to the definition of symmetric matrix, the inverse matrix of A is also symmetric.

Extended data

In the above topic, the properties of the inverse matrix used are: the transposed matrix of the reversible matrix A is also reversible, and the inverse of the transposed matrix is equal to the transposed inverse.

The product of two symmetric matrices is symmetric if and only if their products are commutative. The multiplication of two real symmetric matrices is commutative if and only if their characteristic spaces are the same.

Every real square matrix can be written as the product of two real symmetric matrices, and every composite matrix can be written as the product of two Complex Symmetric Matrix. A matrix is symmetric and obliquely symmetric if and only if all elements are zero.

Other operational properties of transposed matrix:

That is, the determinant of the transposed matrix remains unchanged.

Resources Baidu Encyclopedia-Inverse Matrix