If an nxn matrix has rank n how do you know it's invertible?

In summary, if an nxn matrix has rank n, it is invertible according to the Fundamental Theorem of Invertible Matrices. However, a proof may be necessary and further guidance is needed to approach it. Additionally, if rank(A)=n, dim(Nul(A))=0 and all rows are linearly independent, leading to a non-zero determinant. Furthermore, if Col(A) and Span(A) have the same dimension, this implies that Col(A) is a basis for Span(A). This can only happen when A is invertible.
  • #1
LeakyFrog
22
0
Homework Statement
If an nxn matrix has rank n how do you know it's invertible?

The attempt at a solution
I know that by the Fundamental Theorem of Invertible Matrices if Rank(A) = n then A is invertible. However, I don't know if that is enough of an answer so it kind of seems like I'm supposed to prove it but I'm not sure where to start with that.

Any nudge in the right direction to get my brain cranking is appreciated.
 
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  • #2
If rank(A)=n, what does that say about dim(Nul(A))?
 
  • #3
If an nxn matrix has rank n, then all the rows are linearly independent from one another, so what can you say about it's determinant?
 
  • #4
Let A be that matrix

Col(A) = Span(A)

dim(Col(A)) = dim(Span(A))

What dose that mean? When can the above happen, ever?
 

1. What is the definition of rank for a matrix?

The rank of a matrix refers to the maximum number of linearly independent rows or columns in the matrix. In other words, it is the number of rows or columns that are not linear combinations of other rows or columns.

2. Can a matrix have a rank higher than its dimensions?

No, the rank of a matrix cannot be higher than the number of rows or columns in the matrix. If a matrix has a rank equal to its dimensions, it is considered a full rank matrix and is invertible.

3. How does the rank of a matrix relate to its invertibility?

A matrix with a rank equal to its dimensions is always invertible. This means that all of its rows and columns are linearly independent, making it possible to find a unique solution for every equation in the matrix.

4. Can a matrix with a rank less than its dimensions still be invertible?

No, if a matrix has a rank less than its dimensions, it is considered a singular matrix and is not invertible. This means that there are not enough linearly independent rows or columns to find a unique solution for every equation in the matrix.

5. What is the relationship between the null space and the rank of a matrix?

The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in a zero vector. The dimension of the null space is equal to the number of linearly independent rows or columns that are not included in the rank of the matrix. In other words, the null space and the rank of a matrix are complementary to each other.

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