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

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In summary, the conversation discusses the concept of invertible matrices and whether having a right inverse is sufficient to show that a matrix is invertible. The participants also mention the importance of proving both a right and left inverse for a matrix, and suggest using determinants to do so. They also note that the definition of invertibility is equivalent to the condition that the determinant of the matrix is non-zero. Finally, they acknowledge the relevance of the given information about the dimensions and properties of the matrices involved in the conversation.
  • #1
Yankel
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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 ?
 
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  • #2
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

$\chi$ $\sigma$
 
  • #3
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.
 
  • #4
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 $\det(A)\ne0.$
 
  • #5
Right, so in order to show that A is invertible, I MUST show that is has both a right inverse AND a left inverse ?
 
  • #6
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.
 

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

What does it mean for a matrix to be invertible?

An invertible matrix is a square matrix that has an inverse, meaning that it can be multiplied by another matrix to get the identity matrix. In simpler terms, an invertible matrix is one that can be "undone" with another matrix.

How do I determine if a matrix is invertible?

To determine if a matrix is invertible, you can calculate its determinant. If the determinant is not equal to zero, then the matrix is invertible. Another method is to use the row reduction method to check if the matrix has linearly independent rows or columns.

Can all square matrices be inverted?

No, not all square matrices are invertible. A matrix must satisfy certain conditions, such as having a non-zero determinant, in order to be invertible.

What is the importance of invertible matrices in mathematics?

Invertible matrices are important in many mathematical applications, such as solving systems of linear equations, calculating determinants, and finding the inverse of a transformation. They also play a crucial role in areas of mathematics such as linear algebra, differential equations, and calculus.

How do I find the inverse of a matrix?

To find the inverse of a matrix, you can use the formula: inverse(A) = 1/det(A) * adj(A), where det(A) is the determinant of the matrix and adj(A) is the adjugate matrix (transpose of the cofactor matrix). Alternatively, you can use the row reduction method to find the inverse by transforming the given matrix into the identity matrix.

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