Equivalent Conditions for Invertible nxn Matrices

  • Thread starter eyehategod
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In summary: So you're saying if A transpose is invertible,In summary, the possible equivalent conditions for "A is invertible" are: 1) A is nxn 2) A is invertible matrix theorem 3) det(A) =/= 0 4) rank(A) = n 5) Ax=0 has only the trivial solution 6) Columns of A are linearly independent 7) x -> Ax is 1:1 8) Ax=b has at least 1 solution per b in Rn 9) Columns of A span Rn 10) x -> Ax maps Rn onto Rn 11) CA = I 12) AD = I 13) At is invertible,
  • #1
eyehategod
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write all possible equivalent conditions to "A is invertible," where A is an nxn matrix.
 
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  • #2
...
 
  • #3
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.
 
  • #4
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)
 
  • #5
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 independant
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
 
  • #6
What do you mean by "At is invertible"? What is t?
 
  • #7
HallsofIvy said:
What do you mean by "At is invertible"? What is t?

sorry, the t is for A transform, A^t
 
  • #8
You mean transpose.
 

1. What is an invertible condition?

An invertible condition is a scientific concept that refers to a set of circumstances or factors that, when combined, lead to a specific outcome or result. It is a condition that can be reversed or changed to produce a different outcome.

2. What is the importance of studying invertible conditions?

Studying invertible conditions allows scientists to better understand the underlying mechanisms and factors that contribute to a certain outcome. It also helps in identifying potential solutions or interventions that could change the outcome or prevent it from occurring.

3. How do scientists identify invertible conditions?

Scientists use various methods, such as experiments, observations, and data analysis, to identify and study invertible conditions. They may also use statistical models and simulations to understand the relationships between different factors and outcomes.

4. Can an invertible condition be predicted or controlled?

In some cases, an invertible condition can be predicted and controlled through careful experimentation and data analysis. However, in other cases, there may be multiple factors at play, making it difficult to accurately predict or control the outcome.

5. How do invertible conditions differ from non-invertible conditions?

Invertible conditions can be reversed or changed to produce a different outcome, while non-invertible conditions cannot. Non-invertible conditions often have a single cause or factor that leads to a specific outcome, while invertible conditions may have multiple contributing factors.

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