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Theory: matrices A and B are equivalent if and only if r(A)=r(B).

The rank of matrix is an important numerical character of matrix. 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 ：.

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Theory: matrices A and B are equivalent if and only if r(A)=r(B).

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  1. The rank of matrix is an important numerical character of matrix. 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: Ais a non-degenerate matrix, then the normal form of A is an identity matrix Ewith the same size

  2. Corollary 1: the following propositions are equivalent: Prove:

  3. 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

  4. 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 Ais invertible. （1）The inverse of matrix is unique. Denote the inverse of Aas Let B,C are all inverses of A, then B=EB=(CA)B=C(AB)=CE=C （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?

  5. Review：adjoint matrix Adjoint matrix ?? The order of algebraic cominor! The adjoint matrix of 2-order matrix A . Do you remember?

  6. Formula : It’s an important formula.

  7. Theory: An n-order square matrix Ais invertible if and only if Prove: Keep in mind!

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

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

  10. Properties of the Inverse Keep them in mind！

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

  12. Ex

  13. Method 3: use the definition. Guest：

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

  15. Applications of the inverse—— to solve matrix equations.

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

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