"Linear Algebra and its Applications"-inverse of the matrix

Inverse of the matrix:

Definition of Inverse matrix:

We go back to the inverse of a number when we are studying real numbers, corresponding, in matrix operations, when AB = I, A and B are mutually referred to as inverse matrices, where I is analogous to 1 in real numbers, representing the unit matrix, that is, the diagonal is a matrix of n x N of 1 where the remaining position is 0.

Uniqueness of the inverse matrix:

is the inverse matrix the only thing that exists like the reciprocal of a real number? We may as well prove it briefly. Suppose that the two inverse matrix of a is b,c. According to the definition we have ab=i,ac=i, combined with the basic matrix algorithm, easy to see b=c=ia^-1, thus can see the inverse matrix is the only existence.

How to solve the inverse matrix:

How to solve the inverse matrix problem can be divided into two parts, in the solution of the 2-order matrix, there is a simple algorithm but its proof to be based on the adjoint matrix, and with the increase of the Matrix order is no longer used, so here is not introduced this method in the later introduction of the determinant of the time will give a detailed proof.

The other part is the general algorithm used when solving the inverse matrix of 3 order and above.

First we give a lemma:

Theorem 1: if n x n matrix A is reversible, then for any r^n vector B, the solution to the matrix equation ax = b is only present.

Proof: existence, in this matrix equation is multiplied by the inverse matrix of a, then there is x = a^-1 B. Uniqueness, combined with the properties of the inverse matrix uniqueness mentioned above, prove it.

So now the theoretical basis for the algorithm of the inverse matrix is given:

Theorem 2:n x n matrices are reversible, and when a is equivalent to I, and the elementary line changes experienced in the process of a to I are applied to I, the a^-1 is obtained.

The certification process is as follows.

Based on theorem 2, it is easy to get a general algorithm for solving the inverse matrix of n x n matrices, and then the description of the algorithm is given, and then an example is used to realize the realization of the algorithm.

"Linear Algebra and its Applications"-inverse of the matrix