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orthogonal matrices

Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

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Hi

An inverse of A is matrix A^-1 such that
A A^-1 = A^-1 A = Id.

In this entry, does it follow from Q^T Q = Id
that Q^T = Q^-1? Or more generally, is it possible that
A B = Id for some matrices A, B, but B A \neq Id?

Matte

> is it possible that A B = Id for some matrices A, B,
> but B A \neq Id?

If A and B are squre matrices, then it is impossible I think. Because, consider

A B= id

take det() and you get

det(A) det(B) = 1

so det(A) and det(B) are unequal to zero, so invertable, and so

from A B = id follows that B = A^-1 due to due to the uniqueness of inverse matrix and thus B A = id.

Now, if you have non-square matrices then, first of all

> is it possible that A B = Id for some matrices A, B,
> but B A \neq Id?

there are different Id. And it is fairly possible that

A B = Id_1 but B A \neq Id_2,

example:

(1 0 0)
(0 1 0)
x
(1 0)
(0 1)
(0 0)
=
(1 0)
(0 1)

BUT
(1 0)
(0 1)
(0 0)
x
(1 0 0)
(0 1 0)
\neq
(1 0 0)
(0 1 0)
(0 0 1).

Serg.
-------------------------------
knowledge can become a science
only with a help of mathematics

thanks!

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