- #1
KristenSmith
- 2
- 0
Homework Statement
Show that if A is an n x n matrix whose kth row is the same as the kth row of In, then 1 is an eigenvalue of A.
KristenSmith said:Homework Statement
Show that if A is an n x n matrix whose kth row is the same as the kth row of In, then 1 is an eigenvalue of A.
Homework Equations
None that I know of.
The Attempt at a Solution
I tried creating an arbitrary matrix A and set it up to to find the det(A-λI) but that got me no where.
It is easy to find a specific vector x with Atx = 1x. That shows that x is an eigenvector of At with eigenvalue 1 => At has 1 as eigenvalue => A has 1 as eigenvalue.KristenSmith said:I'm still not understanding what to do with that...
An n x n matrix is a rectangular array of numbers or variables arranged in rows and columns with n rows and n columns. The size of a matrix is typically written as "n x n" or "n by n".
A matrix being "n x n" means that it has an equal number of rows and columns. This is also known as a square matrix.
The significance of A being an n x n matrix is that it allows for certain operations to be performed on it, such as finding the determinant, inverse, and eigenvalues. It also has certain properties and characteristics that are unique to square matrices.
To show that A is an n x n matrix, you can provide the dimensions of the matrix, as well as its entries. For example, you can write A = [aij] where aij represents the entry in the ith row and jth column of the matrix. You can also visually represent the matrix using a grid with n rows and n columns.
A can be any type of matrix as long as it follows the criteria of being an n x n matrix. This means that it has to have an equal number of rows and columns. However, certain operations may only be applicable to specific types of matrices (e.g. square matrices for finding the determinant).