AAt Invertibility: Implications for A Inverse

[JosephR]

Homework Statement

Show that if AAt(where At is the transpose of A) has no inverse,Then A itself cannot have an inverse

2. The attempt at a solution

Here's what i did

AAt.(AAt)-1=AAt.At-1A-1=AA-1=I

thus AAt has an inverse IF and only IF A is invertible and since AAT has no inverse then A has no inverse ...

i thinkt it's somehow wrong the teacher told me that i assumed that AAt is invertible but it's given without an inverse ..any suggestions would be appreciated :)

 

[cristo]

Homework Statement

Show that if AAt(where At is the transpose of A) has no inverse,Then A itself cannot have an inverse

2. The attempt at a solution

Here's what i did

AAt.(AAt)-1=AAt.At-1A-1=AA-1=I

thus AAt has an inverse IF and only IF A is invertible and since AAT has no inverse then A has no inverse ...

i thinkt it's somehow wrong the teacher told me that i assumed that AAt is invertible but it's given without an inverse ..any suggestions would be appreciated :)

You've assumed it's invertible by writing down (AA^T)^-1

Hint: Consider determinants...

 

[JosephR]

i haven't taken determinant till now.. is there any other way to do it ?

 

[HallsofIvy]

You want to prove "If AA^T does NOT have an inverse, then A itself does NOT have an inverse". With all those "not"s, that's what I would call a "negative" statement. It is almost easier to prove the contraposative of a "negative" statement than the statement itself because then you can make a "positive" statement.

Here the contrapositive is "If A has an inverse then so does AA^T."

 

[JosephR]

you mean i should assume that A is invertible and prove that AA^T is invertible also ...