This is the reverse order principle of linear algebra matrix transformation. Just like the transposition of a matrix, the order of the original matrix needs to be reversed. Proof by reverse deduction below:
(1) Perform certification conversion. If the inverse matrix of the AB matrix is required, then the inverse matrix needs to be multiplied by the AB matrix to equal the identity matrix E.
(2) The calculation process is shown in Figure
(3) The discussion is proved
Matrix operations are very different from algebraic operations. When performing matrix assignment operations and squaring operations, the order of the matrix cannot be reversed. Both the inverse matrix and the transposed matrix must satisfy the principle of matrix reverse order.
Extended information:
Let A be an n-order matrix on the number field, if there is another n-order matrix B in the same number field, such that: AB=BA=E, then we call B the inverse matrix of A, and A is called Invertible matrix. Note: E is the identity matrix.
(1) Elementary transformation method for inverse matrix:
Write an n-th order invertible matrix A and n-th order unit matrix I as an nX2n matrix. Perform
elementary row transformation on B, that is, perform identical elementary row transformations on A and I. The goal is to transform A into an identity matrix. When A is transformed into the identity matrix I, the right half of B is transformed into the inverse matrix of A at the same time.
Such as seeking
the inverse matrix A-1.
A reversible and therefore, can be obtained by the right half of the A-1 = inverse matrix
calculation principle elementary transformation method:
If the n-order square matrix A reversible, i.e., line A is equivalent to I, i.e., by the elementary matrices P1, P2, ..., Pk such that
the The two ends of this formula are multiplied by A-1 at the same time:
comparing the two formulas, we can see that the same elementary row transformations are performed on A and I. When these elementary row changes turn A into an identity matrix, these elementary row transformations are also Convert the identity matrix to A-1. [2]
If the matrices A and B are inverse to each other, then AB=BA=I. From the condition AB=BA and the definition of matrix multiplication, we can see that both matrices A and B are square matrices. From the condition AB=I and the theorem "The determinant of the product of two matrices is equal to the product of the determinants of these two matrices", it can be known that the determinants of these two matrices are not zero.
In other words, the rank of these two matrices is equal to their series (or called rank, that is, both A and B are square matrices, and rank(A) = rank(B) = n). In other words, these two matrices can be transformed into identity matrices only through elementary row transformation or elementary column transformation.