Finding 2 constants given two matrices

Question

if A=(3   4) and B =(4   6).   

       (2  -3)             (3 -5)

 

(the brackets above represent one matrix, im just not sure how to write a matrix on this website)

 

and pA +qB = I (where I is the identity)

find the constants q and p 

Answer 1

p(3   4)    +    q(4   6)   =  (1 0)        

  (2  -3)            (3 -5)       (0 1)

 

(p3   p4)    +      (q4   q6)   =  ( p3+q4   p4+q6)       
(p2  -p3)            (q3 -q5)        (p2+q3    -3p-q5)

 

p3+q4 = 1

p4+q6 = 0

p2+q3 = 0

-p3-q5 = 1

 

add the first to the last

 

p3+q4 = 1

-p3-q5 = 1

0 -q = 2

then q = -2

 

plug that in and you get ;

p = 3

Answer 2

The Identity matrix I has top row 10 and bottom row 01, so you have to solve these equations:

1 = 3p +4q

0 = 4p + 6q  

 

0 = 2p +3q

1 = -3p - 5q

 

This is four equations in two unknowns.  Pick any two and solve that system.

Then check that the (p,q) that you obtain also satisfies the other two equations (if not, the problem is not solvable).