Rank and inverse of matrices

In summary, the conversation discusses two linear algebra problems. The first one proves that if B and C are matrices of specific dimensions, then BC has a maximum rank of 1. The second problem shows that if the rank of a matrix is m, then there exists another matrix B such that AB = I. The conversation also includes a discussion on how to approach these problems and how they can be generalized to mxn matrices.
  • #1
ak416
122
0
I have some more linear algebra problems...

First: Prove that if B is a 3x1 matrix and C is a 1x3 matrix, then the 3x3 matrix BC has rank at most 1. Conversely, show that if A is any 3x3 matrix having rank 1, then there exist a 3x1 matrix B and a 1x3 matrix C such that A=BC

The first part is easy (it follows from a theorem). I am not sure how to do the "Conversely" part, and I am also curious about whether it generalizes to mxn matrices and what the linear transformation analogy to this would be.

Second: Let A be an mxn matrix with rank m. Prove that there exists an nxm matrix B s.t AB = I
Also, let B be an nxm matrix with rank m. Prove that there exists an mxn matrix A such that AB = I

Im not sure about these. I know that since the rank is m, there are m linearly independent rows...

Any help would be useful. Thanks.
 
Last edited:
Physics news on Phys.org
  • #2
For the first one, just to reiterate what you've already said, since the rank is one, there is one linearly independent row. So the other two rows are linear combinations of this row. What does that mean?
 
  • #3
Oh ok, i got it. the matrix B could be any of the columns of A. And the matrix C could contain in each corresponding column, the scalar multiple needed to create the corresponding column of A. Thanks. I can see how it would work with nxn, nx1 and 1xn matrices, but with mxn matrices i guess there's something different and more complicated.

Still can't get the second part...
 
  • #4
If there are m linearly independent vectors in Rm, these must span the space, and so any vector can be written as a linear combination of them. Does this help?
 
  • #5
Ok i think i have it. Each column in AB must be a linear combination of the columns of A with the coefficients being the appropriate column in B. By choosing appropriate values for the entries in B, any column vector can be generated for AB (element of R^m). Choose the B entries so that the jth column of AB has zeroes everywhere except at the jth spot a 1. And its similar for the other part of this question. Right?
 
  • #6
That sounds about right.
 

1. What is the rank of a matrix?

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It can also be thought of as the number of non-zero rows or columns in the reduced row echelon form of the matrix. The rank of a matrix is an important concept in linear algebra and is used to determine properties such as invertibility and solutions to systems of equations.

2. How is the rank of a matrix calculated?

The rank of a matrix can be calculated using various methods such as row reduction, Gaussian elimination, or the determinant. The most common method is row reduction, where the matrix is transformed into its reduced row echelon form and the number of non-zero rows is counted.

3. What is the inverse of a matrix?

The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. In other words, it is a matrix that "undoes" the original matrix. The inverse of a matrix is denoted by adding a superscript -1 to the original matrix, e.g. A-1.

4. How is the inverse of a matrix calculated?

The inverse of a matrix can be calculated using various methods such as the Gauss-Jordan elimination method or the adjugate method. The most common method is the Gauss-Jordan elimination method, where the original matrix is augmented with the identity matrix and row operations are performed until the left side becomes the identity matrix, resulting in the inverse on the right side.

5. Can every matrix have an inverse?

No, not every matrix has an inverse. For a matrix to have an inverse, it must be a square matrix (equal number of rows and columns) and also be non-singular (have a non-zero determinant). A matrix that does not have an inverse is called a singular matrix.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
267
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
  • Calculus and Beyond Homework Help
Replies
25
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
2
Replies
40
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
615
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
6K
  • Calculus and Beyond Homework Help
Replies
3
Views
515
Back
Top