Finding the inverse of this matrix.

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SUMMARY

The discussion focuses on finding the inverse of a matrix involving the expression 1n1n', where 1n represents an n x 1 column vector of ones. Participants emphasize the importance of multiplying matrices correctly to demonstrate that the product yields the identity matrix. A key point raised is the need to prove that the product 1n1n' * 1n1n' equals n * 1n1n' through induction. Additionally, clarification is provided regarding the dimensions of the matrices involved, confirming that 1n must be treated as a column vector to maintain dimensional consistency.

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Homework Statement



Hi there I'm trying to solve this question:

dPs5M.png


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?
 
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Kuma said:

Homework Statement



Hi there I'm trying to solve this question:

dPs5M.png


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 1n1n' expressions represent n x n matrices whose entries are all 1's.

The product 1n1n' * 1n1n' works out to be n * 1n1n', which you might need to prove by induction.
 
It says that 1n is a vector of 1's so shouldn't 11' = n?
 
Kuma said:
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.
 
I agree with micromass. 1n has to be a column vector.
 

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