Finding the inverse of this matrix.

[Kuma]

Homework Statement

Hi there I'm trying to solve this question:

Homework Equations

The Attempt at a Solution

I figured i should just multiply them together and show that you get the identity matrix, but I'm having trouble cancelling out some of the terms. I'm not sure if I should write them out in matrix form first or just do them as is?

 

[Mark44]

Homework Statement

Hi there I'm trying to solve this question:

Homework Equations

The Attempt at a Solution

I figured i should just multiply them together and show that you get the identity matrix, but I'm having trouble cancelling out some of the terms. I'm not sure if I should write them out in matrix form first or just do them as is?

I haven't worked this all the way through, but your idea of multiplying the two expressions seems like the way to go.

Here are a couple of tips that might be helpful. The 1_n1_n' expressions represent n x n matrices whose entries are all 1's.

The product 1_n1_n' * 1_n1_n' works out to be n * 1_n1_n', which you might need to prove by induction.

 

[Kuma]

It says that 1n is a vector of 1's so shouldn't 11' = n?

 

[micromass]

It says that 1n is a vector of 1's so shouldn't 11' = n?

They probably mean that $1_n$ is a column vector. Otherwise the dimensions wouldn't agree. Indeed: $(1-\rho)I$ would be a matrix and $1_n1_n^\prime$ would be a number, so you can't add them.

 

[Mark44]

I agree with micromass. 1_n has to be a column vector.