CHAPTER 05.18: SYSTEM OF
EQUATIONS: Some statements about the inverse of matrices Let’s
talk about some statements about inverse of matrices. If A and B are two N by
N matrices- we are talking about square matrices- such that B times A is
equal to I and then we can also say these things also: We can say B is
inverse of A. We can say A is inverse of B. That A and B are invertible; that
means they can be inverted or you can find the inverse. And also we can say,
A times B is equal to I. Although we are able to say hey, B times A is equal
to I then, B is the inverse of A or A is inverse of B, but it also means that
A times B will be equal to the anti-matrix. We can also say that A and B are
nonsingular. And then we can also say that all columns of A and B are
linearly independent. So if we take each-each of the columns of A or B, as
vectors, they will be linearly independent. And so is it true for rows. All
rows of A and B are linearly independent. So these are some things to think
about when you are talking about the inverse of a matrix: to be able to say
hey, if B times A is the anti-matrix and B is inverse of A and A is inverse
of B, A and B are invertible, A times B will be the anti-matrix, both A and B
are considered to be nonsingular, that the columns of A and B are linearly independent,
so are the rows of A and B, which are linearly independent. And that is the
end of this segment. |