Equivalent Conditions for Invertible nxn Matrices

[eyehategod]

write all possible equivalent conditions to "A is invertible," where A is an nxn matrix.

 

[morphism]

...

 

[DeadWolfe]

Well, there's obviously infinitely many, as one can take take any logical statement and compose with a tautology to get a new statement which is true exactly when the original one was.

That said, the only two ones anybody cares to know are the definition, and the requirement that det(A) have an inverse.

 

[daniel_i_l]

Some helpful ones other than the definition(but not all of them!):
1) det(A) =/= 0
2) rank(A) = n (A is an nxn matrix)

 

[Ai52487963]

I'm actually on my way to memorizing the invertible matrix theorem, given I have an exam on it friday.

if A is nxn, then A

a) is invertible
b) ~ I
c) has n pivots
d) Ax=0 has only the trivial solution
e) columns of A are linearly independent
f) x -> Ax is 1:1
g) Ax=b has at least 1 solution per b in Rn
h) columns of A span Rn
i) x -> Ax maps Rn onto Rn
j) CA = I
k) AD = I
j) At is invertible

hope that works

 

[HallsofIvy]

What do you mean by "At is invertible"? What is t?

 

[Ai52487963]

What do you mean by "At is invertible"? What is t?

sorry, the t is for A transform, A^t

 

[morphism]

You mean transpose.