For any two elements A and B that form AB, neither A nor B have to be

In summary, we discussed the concept of invertibility for matrices and how it relates to the existence of an inverse. We also explored the conditions for the inverse of a product of matrices and provided a counterexample. Furthermore, we clarified that for non-square matrices, there can be left and right inverses that are not equal. Finally, we considered the implications of this in the context of linear systems and discussed alternative methods for isolating x.
  • #1
Gear300
1,213
9
For any two elements A and B that form AB, neither A nor B have to be invertible for (AB)-1 to exist, right?
 
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  • #2


Both have to be invertible. Inv(AB) = inv(B)inv(A).
 
  • #3


Right. For example..
[tex] \left[
\begin{array}{c c c}
1&0&0\\
0&1&0
\end{array}
\right]
\left[
\begin{array}{c c}
1&0\\
0&1\\
0&0
\end{array}
\right]~=~\left[
\begin{array}{c c}
1&0\\
0&1\\
\end{array}
\right]
[/tex]

The matrix on the right is clearly invertible, while the two matrices in the product aren't event square, let alone invertible.
 
  • #4


defunc said:
Both have to be invertible. Inv(AB) = inv(B)inv(A).
See my counterexample.
 
  • #5


Thanks for the reply. So that would mean that Inv(AB) = inv(B)inv(A) iff inv(B) and inv(A) exist (whereas Inv(AB) may exist without a defined inv(A) or inv(B)), right?
 
  • #6


Looks correct to me. The condition that I am familiar with is:

If A and B are invertible matrices of the same size, then AB is invertible and (AB)-1 = B-1A-1.

This is easy to prove by showing that (AB)(B-1A-1) = A(BB-1)A-1 = AIA-1 = AA-1 = I.

As far as the other way of your "iff", if (AB)-1 = B-1A-1 is given then it seems to me that the existence of B-1 and A-1 would directly follow since they are used in the initial condition.
 
  • #7


I see. Thanks again for the replies. I have an additional question:

For a matrix C that is not a square matrix, there is no defined inverse; however, it is possible that there is a left inverse A and a right inverse B, in which A =/= B, for the matrix C, right?
 
  • #8


Yes. See the wikipedia article here, under the section titled Matrices.
 
  • #9


Thanks for the link.

So in the case of linear systems Ax = b, I suppose it wouldn't always be possible to use the left inverse of A to isolate x as a general method since the left inverse of A does not necessarily exist. How would one isolate x in these linear systems (other than parametrization of x)?
 

1. What does it mean for two elements to form AB?

When two elements, let's say A and B, combine to form a compound called AB, it means that the atoms of A and B have bonded together through a chemical reaction to create a new substance with different properties than the individual elements.

2. Can elements A and B form more than one type of compound?

Yes, elements A and B can form more than one type of compound depending on the conditions under which they react. For example, hydrogen and oxygen can form water (H2O) or hydrogen peroxide (H2O2).

3. Is it necessary for both elements A and B to be present in equal amounts for them to form a compound?

No, it is not necessary for both elements to be present in equal amounts for them to form a compound. The ratio of elements in a compound depends on the chemical formula of the compound. For example, in water (H2O), the ratio of hydrogen atoms to oxygen atoms is 2:1.

4. Can elements A and B form a compound if they have similar properties?

Yes, elements A and B can still form a compound even if they have similar properties. The properties of a compound are different from the properties of its individual elements. For example, sodium (Na) and chlorine (Cl) have very different properties, but when they combine to form sodium chloride (NaCl), the compound has its own unique set of properties.

5. Is it possible for elements A and B to form a compound without undergoing a chemical reaction?

No, a chemical reaction is necessary for elements A and B to form a compound. This means that the atoms of A and B must bond together in a new arrangement to create a new substance with different properties. If no chemical reaction occurs, the elements will remain in their original form.

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