Let A = 1 1 51 0 1 . If B = 1 2 - 1 - 1 A - 1 - 2 1 1 , then the sum of all the elements of the matrix ∑ n = 1 50 B n is equal to

Step-by-step solution

Given, A = 1 1 51 0 1 , B = 1 2 - 1 - 1   A - 1 - 2 1 1 So, B = M A N Now let M = 1 2 - 1 - 1   &   N = - 1 - 2 1 1 Now solving, 1 2 - 1 - 1 M   - 1 - 2 1 1 N = 1 0 0 1 ∴   M N = I = N M Now using, B = M A N we get B n = M A N n = M A N   M A N   . . . . . . . M A N   ⇒ B n = M A N M ⏟ I A N M ⏟ I   . . . . . . . N M ⏟ I A N   ⇒ B n = M A A A A A . . . . . A ⏟ n   times N   = M A n N Now, using A = 1 1 51 0 1 &