What is the rank of an nxn matrix and how is it determined?

In summary, the conversation discusses a problem from Applied Linear Algebra by Olver, namely finding the rank of an nxn matrix. The equation Rank(A) = the number of pivots in Matrix A is mentioned. The person has attempted to solve the problem but feels stuck and asks for guidance. Another person suggests checking the number of linearly independent columns or zero rows in the reduced matrix. After some clarification, the person realizes their mistake in reducing the matrix and provides a solution.
  • #1
JonathanT
18
0

Homework Statement


http://img94.imageshack.us/img94/5227/nxnmatrix.png

Homework Equations



Rank(A) = the number of pivots in Matrix A.

The Attempt at a Solution



I've spent some time rewriting the matrix and other operations. I really just feel like I'm banging my head against the wall. Not making any progress. Just need to be pointed in the right direction. Oh and the answer is "2." No idea how they get that. Thanks for any help in advance.

Edit: For reference this problem is from Applied Linear Algebra by Olver. Problem 1.8.17.
 
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  • #2
JonathanT said:

Homework Statement


http://img94.imageshack.us/img94/5227/nxnmatrix.png


Homework Equations



Rank(A) = the number of pivots in Matrix A.

The Attempt at a Solution



I've spent some time rewriting the matrix and other operations. I really just feel like I'm banging my head against the wall. Not making any progress. Just need to be pointed in the right direction. Oh and the answer is "2." No idea how they get that. Thanks for any help in advance.

Stop and ask yourself how many linearly independent columns compose your matrix A.

If that notion confuses you, ask yourself how many zero ROWS you have after row reduction...
 
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  • #3
I think I get it. I was reducing the matrix wrong. I think I have the correct rref of the matrix now. Take a look and let me know what you think.

Solution
 

1. What is the rank of an nxn matrix?

The rank of an nxn matrix is the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the vector space spanned by the rows or columns of the matrix.

2. How is the rank of an nxn matrix calculated?

The rank of an nxn matrix can be calculated by performing row reduction on the matrix and counting the number of non-zero rows in the reduced matrix. Alternatively, the rank can also be determined by finding the number of non-zero eigenvalues of the matrix.

3. What is the significance of the rank of an nxn matrix?

The rank of an nxn matrix is significant because it provides information about the linear independence of the rows and columns of the matrix. It also determines the dimension of the vector space spanned by the matrix, which is important in various applications of linear algebra.

4. Can the rank of an nxn matrix be greater than n?

No, the rank of an nxn matrix cannot be greater than n. This is because there can only be a maximum of n linearly independent rows or columns in an nxn matrix.

5. How does the rank of an nxn matrix affect its inverse?

The rank of an nxn matrix plays a crucial role in determining whether the matrix is invertible or not. If the rank of the matrix is less than n, then the matrix is not invertible. However, if the rank is equal to n, then the matrix is invertible and its inverse can be calculated using various methods such as Gauss-Jordan elimination.

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