If A is 2x2 matrix ( a b c d ) and A-1 is ( x y z w ) . Find w x y z in the term of a b c d.

Question

If A= (a b) and A-1= ( x y )

         ( c d)                 ( z w)

Find w,x,y,z in the terms of a,b,c and d.

 

P/s: I really appreciate if you can help me. Thank you sir/madam/miss. 

Answer 1

Student,

 

You can find the components of A^-1 in terms of the components of A from

 

A^-1 = [1/det(A)] adj(A),

 

where adj(A) is the transpose of the matrix where each element of A is replaced by its cofactor, i.e. adj(A) is the transpose of the matrix of the minors, where each minor is multiplied by (-1)^i+j.  It is tedious to type all of this in this Answer window, but you can look up the relations in your linear algebra textbook.  

Answer 2

You can use the property: AA^-1=E (E-identical matrix):

If you multiply Aand A^-1 you will obtain the following system of  equations:

 

ax+bz=1

ay+bw=0

cx+dz=0

cy+dw=1

 

Solve this system for x, y, z and w:

x=d/(ad-bc), y=b/(bc-ad), z=z/(ad-bc), w=a/(ad-bc).