Finding the inverse of this matrix.

In summary: In that case, 1n1n' is an n x n matrix, so 1n1n' * 1n1n' is an n x n matrix. The product 1n1n' * 1n1n' works out to be n * 1n1n', which you might need to prove by induction.In summary, the conversation discusses solving a question involving multiplying two expressions to get the identity matrix. The suggestion is to multiply 1n1n' * 1n1n' and use induction to prove the solution. There is also discussion about the dimensions of 1n and 1n1n' and the need for 1n to be a column vector
  • #1
Kuma
134
0

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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  • #2
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.
 
  • #3
It says that 1n is a vector of 1's so shouldn't 11' = n?
 
  • #4
Kuma said:
It says that 1n is a vector of 1's so shouldn't 11' = n?

They probably mean that [itex]1_n[/itex] is a column vector. Otherwise the dimensions wouldn't agree. Indeed: [itex](1-\rho)I[/itex] would be a matrix and [itex]1_n1_n^\prime[/itex] would be a number, so you can't add them.
 
  • #5
I agree with micromass. 1n has to be a column vector.
 

FAQ: Finding the inverse of this matrix.

What is the inverse of a matrix?

The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. It is denoted by A-1 and is only defined for square matrices.

How do you find the inverse of a matrix?

To find the inverse of a matrix, you can use the Gauss-Jordan elimination method or the adjugate matrix method. Both methods involve performing operations on the original matrix to transform it into its inverse.

Why is finding the inverse of a matrix important?

Finding the inverse of a matrix is important in many areas of mathematics and science, such as solving systems of linear equations, computing determinants, and solving differential equations. It also has practical applications in fields such as engineering, physics, and economics.

Can every matrix be inverted?

No, not every matrix has an inverse. A matrix must be square (i.e. have the same number of rows and columns) and have a non-zero determinant in order to have an inverse. If the determinant is zero, the matrix is said to be singular and cannot be inverted.

Is the inverse of a matrix unique?

Yes, the inverse of a matrix is unique. This means that there is only one possible inverse for a given matrix. If a matrix has an inverse, it is the only matrix that will result in the identity matrix when multiplied by the original matrix.

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