Is Matrix A Invertible Given (AB)C Equals the Identity Matrix?

  • Context: MHB 
  • Thread starter Thread starter Yankel
  • Start date Start date
  • Tags Tags
    Matrix
Click For Summary

Discussion Overview

The discussion revolves around the question of whether the equation (AB)C = I implies that matrix A is invertible. Participants explore the implications of this equation in the context of matrix algebra, focusing on the definitions of invertibility and the properties of matrix multiplication.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant states that (AB)C = I can be rewritten as A(BC) = I, suggesting that BC could be considered a right inverse of A.
  • Another participant questions whether showing BC is a right inverse is sufficient, asking if it is necessary to also demonstrate that (BC)A = I.
  • Some participants propose that A could be either invertible or not, indicating uncertainty in proving A's invertibility using algebra alone.
  • It is mentioned that knowing A has a right inverse BC does not automatically imply that BC is a left inverse, and proving this is not straightforward.
  • One participant suggests using determinants to show A is invertible, noting that the condition det(A) ≠ 0 is equivalent to A being invertible.
  • Another participant emphasizes the importance of A being a finite square matrix, as indicated by the definitions and dimensions involved in the discussion.

Areas of Agreement / Disagreement

Participants express differing views on whether the given equation implies A's invertibility, with some asserting it could be invertible while others remain uncertain. There is no consensus on a definitive conclusion regarding A's invertibility.

Contextual Notes

Participants highlight the relevance of matrix dimensions and properties, such as the necessity for A to be a square matrix for determinants to be defined. There are unresolved aspects regarding the proof of A's invertibility based solely on the provided equation.

Yankel
Messages
390
Reaction score
0
Hello all

I am told that (AB)C=I and I am asked if that means A is invertible.
I also knows that A+A^t is defined (irrelevant as far as I understand) and that A has 3 rows, also irrelevant.

what I did is:

(AB)C=A(BC)=I

and then BC is the inverse of A. But according to the definition that ain't sufficient, is it ? Do I need also to show that (BC)A=I ? If so, how ?
 
Physics news on Phys.org
Yankel said:
Hello all

I am told that (AB)C=I and I am asked if that means A is invertible.
I also knows that A+A^t is defined (irrelevant as far as I understand) and that A has 3 rows, also irrelevant.

what I did is:

(AB)C=A(BC)=I

and then BC is the inverse of A. But according to the definition that ain't sufficient, is it ? Do I need also to show that (BC)A=I ? If so, how ?

Because for the product of matrices the associative property holds, it will be ...

$\displaystyle (A \cdot B) \cdot C = I \implies A \cdot (B \cdot C) = I\ (1)$

The commutative property are however not true in general, so that ...

$\displaystyle A \cdot (B \cdot C) \ne (B \cdot C) \cdot A\ (2)$

Kind regards

$\chi$ $\sigma$
 
so it could be invertible and it could be not invertible, I have no way to proof one way or another by using algebra only.
 
You know that A has a right inverse BC, and you want to show that BC is also a left inverse for A. This is true, but not that easy to prove. If you know about determinants, the best way to show that A is invertible is to use the fact that this is equivalent to the condition $\det(A)\ne0.$
 
Right, so in order to show that A is invertible, I MUST show that is has both a right inverse AND a left inverse ?
 
Yankel said:
Right, so in order to show that A is invertible, I MUST show that is has both a right inverse AND a left inverse ?

It suffices to make deductions about the determinants of $A$, $(BC)$, and $I$, using that for any finite square matrices $P$ and $Q$, we have:
$$\det(PQ)=\det P \cdot \det Q$$

Note that it is relevant that $A+A^t$ is defined and that $A$ has 3 rows, since it implies that A is a finite, square matrix.
Furthermore, the fact that $A(BC)=I$, implies that $(BC)$ is also a square matrix of the same dimensions as $A$.
You need this, because otherwise the determinants would not be defined.
 

Similar threads

MHB Is B Equal to A³ Given Symmetric and Invertible Matrices?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Prove that if A is an invertible matrix and AB = BC then B = C.
  • · Replies 1 ·
Replies
1
Views
19K
MHB What is the center of $GL_n(\mathbb{R})$?
  • · Replies 7 ·
Replies
7
Views
2K
Transformations to both sides of a matrix equation
  • · Replies 25 ·
Replies
25
Views
3K
Prove B is invertible if AB = I
  • · Replies 40 ·
2
Replies
40
Views
6K
Is it ok to assume matrices A and B as identity matrix?
  • · Replies 25 ·
Replies
25
Views
2K
Prove the identity matrix is unique
  • · Replies 69 ·
3
Replies
69
Views
10K
Graduate Algebraic proof that Euler angles define a proper rotation matrix
  • · Replies 6 ·
Replies
6
Views
2K
MHB Solving the Matrix Transformation: $B \to C$
  • · Replies 1 ·
Replies
1
Views
1K