Rank of a Matrix: Can We Eliminate Lines to Get Non-Vanishing Determinant?

In summary, if A is an nxk matrix of real numbers (n>=k) of rank k, it is possible to eliminate n-k lines of A to obtain a kxk matrix A' of nonvanishing determinant. This is because the row space of A is spanned by k vectors, so eliminating (n-k) rows will leave A' with a rank of k and make it invertible, ensuring a non-zero determinant.
  • #1
quasar987
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If A is an nxk matrix of real numbers (n>=k) of rank k, is it true that we can eliminate n-k lines of A to obtain a matrix A' of nonvanishing determinant?

I convinced myself of that one time while in the bus and now I can't find the proof.
 
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  • #2
Hmm, if A is of rank k, then that means that the row space of A is spanned by k vectors , and this means that we can eliminate (n-k) rows of A which are effectively linear combinations of the others. So when we do that we have A', which is a kxk matrix and of rank k, which implies that it is invertible which in turn implies its det is non-zero.
 
  • #3
Ohhh.. yeah!

Thanks!
 
  • #4
Welcome. I'm trying to jog my linear algebra memory for a intermediate linear algebra class this semester.
 

FAQ: Rank of a Matrix: Can We Eliminate Lines to Get Non-Vanishing Determinant?

What is the rank of a matrix?

The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix.

How is the rank of a matrix related to its determinant?

The rank of a matrix is related to its determinant because the determinant of a matrix is zero if and only if the rank of the matrix is less than its dimensions.

Can we eliminate lines from a matrix to get a non-vanishing determinant?

Yes, it is possible to eliminate lines from a matrix to get a non-vanishing determinant. This is known as row reduction or Gaussian elimination, and it involves performing elementary row operations on the matrix to transform it into an upper triangular form without changing the determinant.

Why is the rank of a matrix important?

The rank of a matrix is important 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 rows or columns of the matrix.

How is the rank of a matrix calculated?

The rank of a matrix can be calculated by performing row reduction on the matrix and counting the number of non-zero rows in the reduced matrix. Alternatively, it can be calculated by finding the largest square submatrix with a non-zero determinant.

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