Rank of a Matrix: Determine Value of k

In summary, the rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It can be determined by using row reduction techniques to transform the matrix into its reduced row echelon form. The purpose of finding the rank is to determine properties and for various applications. The value of k does not directly affect the rank, but it can be multiplied by a non-zero constant without changing the rank. The rank of a matrix can never be greater than its dimensions, but it can be equal to the number of rows or columns.
  • #1
lalligagger
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Homework Statement



Determine the values of k, if any, that give the matrix (1,1,k),(1,k,1),(k,1,1) a rank of: zero, one, two, or three.

Homework Equations





The Attempt at a Solution



I tried reducing to row echelon form but it's confusing dealing with all the k's. Is there a better approach to this?
 
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  • #2
The row echelon form of this matrix is not very complicated, however it depends on the value of k. So whenever you divide by some value involving k while computing it, you will need to consider special cases (k=1, k=-2).
 

1. 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. It can also be thought of as the number of non-zero rows or columns in the matrix after performing row or column operations.

2. How do you determine the rank of a matrix?

To determine the rank of a matrix, you can use row reduction techniques to transform the matrix into its reduced row echelon form. The number of non-zero rows in the reduced form will be equal to the rank of the original matrix.

3. What is the purpose of finding the rank of a matrix?

The rank of a matrix is useful in determining certain properties of the matrix, such as its invertibility and the number of solutions to a system of linear equations represented by the matrix. It is also used in various applications in fields such as engineering, physics, and economics.

4. How does the value of k affect the rank of a matrix?

The value of k does not directly affect the rank of a matrix. However, it is possible to multiply a row or column of a matrix by a non-zero constant without changing the rank. Therefore, if k is a non-zero constant, it will not affect the rank of the matrix.

5. Can the rank of a matrix be greater than its dimensions?

No, the rank of a matrix can never be greater than its number of rows or columns. This is because the maximum number of linearly independent rows or columns in a matrix is limited by its dimensions. However, the rank can be equal to the number of rows or columns in the matrix if the matrix is a full rank matrix.

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