Dear Rupaj Kumar Nayak,

Let  X = I be the identity matrix of order n,  and Y = diag(d₁,d₂,⋯,d_{n}) be a diagonal matrix of order n.

In fact the inverse of a diagonal matrix is also a diagonal matrix, if d_{i} ≠ 0 and d_{i} ≠ -1 for all i = 1,2,…,n. Then we have

Y⁻¹ = diag(1/d₁,1/d₂,⋯,1/d_{n}),

and

I + Y⁻¹  =  diag(1+(1/(d₁)),1+(1/(d₂)),⋯,1+(1/(d_{n}))

            = diag((d₁+1)/d₁, (d₂+1)/d₂,⋯, (d_{n}+1)/d_{n}).

Therefore

(I +Y⁻¹)⁻¹ = diag(d₁/(d₁+1), d₂/(d₂+1), ⋯, d_{n}/(d_{n}+1)).

Regards,

Wiwat Wanicharpichat

Dear Rupaj Kumar Nayak,

Let  X = I be the identity matrix of order n,  and Y = diag(d₁,d₂,⋯,d_{n}) be a diagonal matrix of order n.

In fact the inverse of a diagonal matrix is also a diagonal matrix, if d_{i} ≠ 0 and d_{i} ≠ -1 for all i = 1,2,…,n. Then we have

Y⁻¹ = diag(1/d₁,1/d₂,⋯,1/d_{n}),

and

I + Y⁻¹  =  diag(1+(1/(d₁)),1+(1/(d₂)),⋯,1+(1/(d_{n}))

            = diag((d₁+1)/d₁, (d₂+1)/d₂,⋯, (d_{n}+1)/d_{n}).

Therefore

(I +Y⁻¹)⁻¹ = diag(d₁/(d₁+1), d₂/(d₂+1), ⋯, d_{n}/(d_{n}+1)).

Regards,

Wiwat Wanicharpichat

I think the verbal expression of your question doesn't reflect that equation you put in your comment.  That is perfectly right according to the your equation but  not for the verbal representation:

(I + (I+Y)^-1) = (I+ [ 1/(1+y k)Delkj ]=  (yk +2)/(1+yk ) Delkj

Where  Delkj is Kronecker Delta function. 

Regards

Let In be the identity matrix of order n, and Dn=diag(d1,d2,d3,.........,dn) be the diagonal matix such that all di's  are non-zeros. Then (In+Dn)-1=In+ (Dn)-1=

  In+diag(1/d1,1/d2,1/d3,.........,1/dn)=diag(1+1/d1,1+1/d2,1+1/d3,.........,1+1/dn). Similarly,  we get the same result for (Dn+In)-1.

What I Feel he is asking to find the inverse of the matrix ( X + Y), Let X = I be the identity matrix of order n, and Y = diag(d₁,d₂,⋯,d_{n}) be a diagonal matrix of order n.

Thus  X + Y =  diag(1+d₁, !+d₂,⋯,1+d_{n}) and so 

inverse of the matrix ( X + Y) = diag(1/(1+d₁),1/(1+d₂),⋯,1/(1+d_{n})).

The first answer is the best)

Dear Rupaj

I agree with Alexandre.

The first answer is perfect.

Regards. Tadeusz

 First of all we have to check whether X+Y is  non singular  or not, If so, then sum is also a diagonal matrix and  hence its inverse is the reciprocals of its diagonal elements. 

The final solution is attached in details

Best Luck