Can Rank of A Determine Rank of A+A²+A³+A⁴?

In summary, the problem of rank of a matrix refers to determining the number of linearly independent rows or columns in a given matrix. It can be calculated by reducing the matrix to its reduced row echelon form and the rank tells us the maximum number of linearly independent rows or columns in the matrix. A matrix can have a rank of zero if all its elements are zero. The rank of a matrix is related to its determinant, but the rank can be lower than the size of the matrix even if the determinant is non-zero.
  • #1
Suvadip
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If rank of A is 2. Is it possible to find the rank of A+A2+A3+A4

from that information? Please help
 
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  • #2
suvadip said:
If rank of A is 2. Is it possible to find the rank of A+A2+A3+A4

from that information? Please help

It can't be determined, denoting $B=A+A^2+A^3+A^4$: $$\left \{ \begin{matrix}A=I\Rightarrow B=4I\Rightarrow\mbox{rank } B=2\\A=\begin{bmatrix}{1}&{\;0}\\{0}&{-1}\end{bmatrix}\Rightarrow B=\begin{bmatrix}{4}&{0}\\{0}&{0}\end{bmatrix} \Rightarrow\mbox{rank } B=1\\A=\begin{bmatrix}{-1}&{\;\;0}\\{\;\;0}&{-1}\end{bmatrix}\Rightarrow B=0\Rightarrow\mbox{rank } B=0\end{matrix}\right. $$
 

FAQ: Can Rank of A Determine Rank of A+A²+A³+A⁴?

What is the problem of rank of a matrix?

The problem of rank of a matrix refers to determining the number of linearly independent rows or columns in a given matrix. It is an important concept in linear algebra and has various applications in fields such as statistics, engineering, and computer science.

How is the rank of a matrix calculated?

The rank of a matrix can be calculated by performing row operations and reducing the matrix to its reduced row echelon form. The number of non-zero rows in the reduced row echelon form is the rank of the original matrix.

What does the rank of a matrix tell us?

The rank of a matrix tells us the maximum number of linearly independent rows or columns in the matrix. It also provides information about the dimension of the vector space spanned by the rows or columns of the matrix.

Can a matrix have a rank of zero?

Yes, a matrix can have a rank of zero if all its elements are equal to zero. This means that all the rows and columns of the matrix are linearly dependent, and there is no information contained in the matrix.

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

The rank of a matrix is related to its determinant in the sense that a matrix with a non-zero determinant will have a non-zero rank. However, the rank of a matrix can be lower than its size, even if its determinant is non-zero. This is because the determinant only considers the linear independence of the rows or columns, while the rank also takes into account the number of non-zero entries in the matrix.

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