(a, e) is a matrix with n rows and 2n columns obtained by placing e on the right side of A. As a block matrix, it has one row and two columns.
P Because the block matrix is a row and a column, according to the block matrix multiplication rule, 【 consistent with the usual matrix multiplication 】:
P(a, e)=(pa, pe), and pa = b,
Pe = p., so p(a, e)=(b, p).
What we are talking about here is how to find the inverse matrix of A by elementary transformation. Let p = a (- 1). Then pa = b = e. The above formula becomes
a^(- 1)(a,e)=(e,a^(- 1))，
A (- 1) is an invertible matrix, which is equal to the product of some elementary matrices. For example, a (- 1) = f 1f2f3.
f 1f2f3(a,e)=(e,a^(- 1))，
Note that a matrix is left multiplied by an "elementary matrix". Therefore, it is different from performing row elementary transformation on the matrix (i.e.,
The elementary transformation results of the rows where e becomes "elementary matrix" are equal.
That is to say, (a, e) elementary transformation is carried out three times. Get (e，a (- 1))，
That is to say, (a, e) is transformed elementary. When A on the left becomes identity matrix E, E on the right becomes.
A (- 1), which is the method of finding the inverse of elementary transformation. It should be noted that.
If a
Irreversible, then a is transformed by lines, and e cannot be changed. It won't work out.
②(a, e) can only be transformed by elementary lines.
③ If
┌a┐
└e┘ Use column elementary transformation. When a becomes e, the latter e becomes a (- 1).