If AB=I, then A is invertible. False

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SUMMARY

The assertion that if AB=I, then A is invertible is false. A counterexample involves a 2x3 matrix A and a 3x2 matrix B, where their product results in the identity matrix I, a 2x2 matrix. In this case, neither A nor B is invertible, demonstrating that the dimensions of the matrices play a crucial role in invertibility. Thus, the condition AB=I does not guarantee the invertibility of A or B.

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If AB=I, then A is invertible.

False, but not sure how to show it.
 
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Dustinsfl said:
If AB=I, then A is invertible.

False, but not sure how to show it.
There's no requirement that A and B have to be square matrices. I came up with a 2x3 matrix and a 3x2 matrix whose product is I, a 2x2 matrix, yet neither of the matrices in the product is invertible.
 
Thanks, I figured out a matrix that works.
 
If A(n*n) and B (n*n).
and If AB=I.
is A invertible?
is B inverse of A?
 
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