2x2 matrices

 where

each element is integer

each inverse element is integer

* Matrix should be non-singular

* Determinant of such a matrix will be +1 or -1.

If a, b are first row elements & c, d are second row elements

then, ad =bc ± 1

* In a general case where the inverse need not be an integer and ad=bc ± x, trD=(trA)2±2x; trB =trA /± x

        
Matrix A a   b   
det: tr: c   d   
eigen value:λ       ±  
ratio of eigenvector components (y/x):       ±  
ratio1: ratio2:    
Matrix B (A-1 ) e   f   
det: tr: g   h  
eigen value:λ       ±  
ratio of eigenvector components (y/x):       ±  
       
Matrix C (A + B ) i   j   
det: tr: k   l  
       
Matrix (A / B) =D or A=BD=DB m   n   
det: tr: o   p  
eigen value:       ±  
ratio of eigenvector components (y/x):       ±  
Similarity  transformation Matrix (for A)  
λ1:  r1(λ1-a): r3(λ1-d):  
λ2:  r2(λ2-a): r4(λ2-d):  
S (transformation matrix) u: v:  
put arbitrary value for u,v w: y:  
w/u= y/v=    
uy/wv = r2/r1=    
det: tr:      
       
S-1 u1: v1:  
  w1: y1:  
det: tr:      
       
       
S-1AS s11 s12  
  s21 s22  
       
       

* to find elements of matrix D, we have the following 4 equn.

dm=aΔ+bo;  dn=bΔ+bp; ao=cΔ+cm; ap=dΔ+cn; where Δ is determinant of A. solving, we get m=a2 +bc; n=b(a+d); o=c(a+d); p= d2 +bc;

*| 3 -7 | is a matrix A where A+B=I, AB=I

  |1  -2|

* A + B = | a±d     b-+b | if Δ=1,it is b-b (c-c) and if Δ=-1 , it is b+b (c+c)

                 | c-+c    a±d  |   and if Δ =1 , it is a+d, and if Δ=-1, it is a-d;

* take any singular matrix, D will be (a+d) times a matrix or trA* a matrix

*tr D =(trA)2±2;

* in a similarity transformation, w/u =(λ1 -a)/b =(d-a)/2b  +(1/b)√[(tr/2)2-determinant];

                                                    y/v= (λ2 -a)/b=(d-a)/2b  - (1/b)√[(tr/2)2-determinant];

which are nothing but the ratio of y/x component of eigen vectors of A.

which means that out of 4 elements of the matrix, 2 are to be arbitrarily specified and the other 2 shall be fixed as per above formula .