Show that if A is invertible and AB AC then B C For A 1 3

Show that if A is invertible and AB = AC, then B = C. For A = [1 3 2 6] come up with two matrices B and C such that AB = AC but B = C.

Solution

a) Since A is invertibel so A^-1 exists:

So, AB = AC

Multiply both sides by A^-1 : A^-1AB = A^-1AC

(AA^-1 =1)

B = C ( Hence proved)

b) A = ( 1 3 , 2 , 6) a b

c d

Let B = ( a b , c d)

Let AB = 0 ---> a +3c =0 ;

2b +6d =0

Look for possible values of a, b, c, d : a = 3 ; b =-1

c = -3 ; c = 1

So, B = ( 3 -1 , -3 1)

we can have C = ( 0 0 , 0 0 ) matrix

So, AB = AC