Prove that each matrix is invertible and find their inverses

Question

I have three $n\times n $ matrices $A,B,C$ and it is given that $ABC=I$, I should prove that each $A,B$ and $C$ are invertible and find their inverse.

Here is what I have: Since $ABC=I$ then I might have that $A$ and $B$ and $C$ are inverses of each other, but from this moment I got confused.

A matrix really has only one inverse. You know $ABC=I$, so how can you solve $AX=I$ for $X$? — Angina Seng, Nov 30 '17 at 20:11
If at least one of the three matrices is not invertible, its deteminant is $0$ implying that the determinant of the product of the matrices is $0$. This contradicts $\det(ABC)=\det(I)=1$ — Peter, Nov 30 '17 at 20:14
Here is a hint to find $B^{-1}\cdots(ABC)^{-1} = C^{-1}B^{-1}A^{-1} = I^{-1} = I$ — Doug M, Nov 30 '17 at 20:17

Riley · Answer 1 · 2017-11-30T20:18:49.657

1

$A$ is invertible because $BC$ is its inverse.

$C$ is invertible because $AB$ is its inverse.

$B=A^{-1}IC^{-1}$, a product of invertible matrices, and is therefore invertible. To find $B^{-1}$, we have $$B^{-1}=(A^{-1}C^{-1})^{-1}=CA$$

edited Nov 30 '17 at 20:18

answered Nov 30 '17 at 20:12

Riley

2,845

score 0 · Answer 2 · answered Nov 30 '17 at 20:13

0

Associativity gives $$I=A(BC)=(AB)C$$these should yield $A^{-1}$ and $C^{-1}$ pretty easily. What do you think we can do for $B^{-1}$ (hint: the previous part says $A$ and $C$ are invertible)?

answered Nov 30 '17 at 20:13

Dave

13,832

Prove that each matrix is invertible and find their inverses

2 Answers2