# Thread: eigenvalues and singular matrices problem

1. ## eigenvalues and singular matrices problem

Consider the following matrices:

A1 = 1 1
1 0 ,

A2 = 1 0
0 0 ,

A3 = 1 1
1 1 ,

A4 = 0 1
0 0 ,

A5 = 1 −1
1 0.

A1,A2,A3,A4,A5 are 2*2 matrix, sorry for the format, I don't know how to type the matrix in computer.

1.Compute eigenvalues of each matrix, repeated according to multiplicity
2.Which of the matrices are singular?
3.For each i, either show that Ai cannot be diagonalized, or give invertible P anddiagonal D such that Ai = (p)(D)(P^(-1))

2. Originally Posted by shannon1111
Consider the following matrices:

A1 = 1 1
1 0 ,

A2 = 1 0
0 0 ,

A3 = 1 1
1 1 ,

A4 = 0 1
0 0 ,

A5 = 1 −1
1 0.

A1,A2,A3,A4,A5 are 2*2 matrix, sorry for the format, I don't know how to type the matrix in computer.

1.Compute eigenvalues of each matrix, repeated according to multiplicity
2.Which of the matrices are singular?
3.For each i, either show that Ai cannot be diagonalized, or give invertible P anddiagonal D such that Ai = (p)(D)(P^(-1))
Okay, well what have you done? Do you know how to find the eigenvalues? For 2 by 2 matrices it is just solving a quadratic equation.

A matrix is singular if and only if it has at least one 0 eigenvalue.

An n by n matrix can be diagonalized if and only if it has n independent eigen[b]vectors/b] in which case the matrix $\displaystyle P^{-1}$ is the matrix having those eigenvectors as columns.