Problem 1

Let A be the matrix whose entries are all equal to 1:

Find its eigenvalues and associated eigenspaces.

Hint: you may use the fact, proven later in the course, that the sum of the dimensions of its eigenspaces may not exceed ).

Our eigenvalues are with eigenvectors where and with vectors where

Our eigenspaces are then

For we can rewrite it as meaning our eigenspace is in dimensions adding this with our 1 dimension eigenspace we get a total sum of dimensions of yielding our hint.

Problem 2

A matrix A is nilpotent if there exists an integer such that . The smallest value of is called the index of .

Let be an nilpotent matrix of index , and let be the identity matrix.

(a). Prove that is not Invertible

Assume is invertible

Since we arrive at a contradiction and can conclude that is not invertible. Since the product of invertible matrices must be invertible we can thus conclude that is also not invertible.

(b). Verify that is Invertible by Finding Its Inverse (Hint: Use the Fact that )

Any matrix multiplied by the identity matrix returns itself so .

is invertible and it's inverse is .

(c). Prove that the only Eigenvalue of is 0

Via (a) we proved that is not invertible we thus know that 0 is an eigenvalue of via the invertible matrix theorem. For a value to be an eigenvalue

0 is the only valid eigenvalue of .