Finding the inverse of this matrix.

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The discussion revolves around finding the inverse of a matrix by multiplying two expressions to demonstrate that the product yields the identity matrix. Participants suggest that the user should write the matrices in standard form to facilitate calculations. There is clarification that the notation 1n represents a column vector of ones, which is crucial for understanding matrix dimensions. Additionally, it is noted that the product of the matrices involves proving certain properties, such as the result of multiplying 1n1n' leading to n times 1n1n'. Overall, the conversation emphasizes the importance of proper notation and understanding matrix operations in solving the problem.
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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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