Obviously, if the ranks of two matrices are equal, they have the same normal forms, and so they are equivalent; conversely, if two matrices are equivalent, their ranks are equal too. That is ： Theory: matrices A and B are equivalent if and only if r(A)=r(B). !!! Please remember:we need to figure out if the ranks of matrices are equal only and we will know if they are equivalent. Non-degenerate Matrix Definition: if the rank of square matrix A is equal to its order, we call A a non-degenerate matrix. Otherwise, degenerate matrix. ( non-degenerate  non-singular; degenerate  singular ） E----non-degenerate matrix O----degenerate matrix Theory: A is a non-degenerate matrix, then the normal form of A is an identity matrix E with the same size The rank of matrix is an important numerical character of matrix.

Corollary 1: the following propositions are equivalent: Prove:

Corollary 2 : Matrices A and B are equivalent if and only if there are m-order and n-order non-Degenerate matrices P,Q, such that And we also have ： If P,Q are non-degenerate, then r(A) = r(PA) = r(PAQ) = r(AQ) e.g. The Inverse of a Matrix

Definition ： if A is an n-order square matrix, and there is another n-order square matrix B such that AB=BA=E, we say that B is an inverse of A, and A is invertible. （ 1 ） The inverse of matrix is unique. Let B,C are all inverses of A, thenB=EB=(CA)B=C(AB)=CE=C Denote the inverse of A as （ 2 ） Not all square matrices are invertible. For example is not invertible. It’s impossible. So A is not invertible. The questions to answer: 1. When the matrix is invertible? 2. How to find the inverse?

Review ： adjoint matrix Adjoint matrix The order of algebraic cominor! The adjoint matrix of 2-order matrix A.

It’s an important formula. Formula :

Theory: An n-order square matrix A is invertible if and only if Prove: Keep in mind!

e.g.1. Solution ： e.g.2. Prove: By the same method, we can prove others

That is, the inverses of elementary matrices are elementary matrices of the same size. ——This is the 3 rd property of elementary matrices 。 Exercises: Find the inverse. ？？ ？ How to find the inverse of

Properties of the Inverse

Methods to Find the Inverse Method 1 ： Method 2 ： Use elementary operations to find the inverse.

Ex

Method 3: use the definition. Guest ：

Method 4: prove B is the inverse of A by definition.

Applications of the inverse—— to solve matrix equations.

When we solve matrix equations, remember that before figuring out the solutions, reduce the matrices at first.