Question
Tue June 05, 2012 By: Prkriti Gupta
 

IF A BE a SQUARE MATRIX SUCH THAT A(SQUARE)=A THEN SHOW THAT (I+A)RAYS TO POWER N=I+[2(RAYS TO POWER N)-1]*A for all natural numbers n

Expert Reply
Tue June 05, 2012

For n = 1, (I+A)1 = I+[21-1]*A = I + A

So, the result is true for n = 1

Let the result be true for n = k. Then, we have:

(I+A)k = I+[2k-1]*A        … (1)

For n = k + 1,

(I+A)k+1 = (I+A)k (I + A)

          = [I+[2k-1]*A] (I + A)             [Using (1)]

          = I+[2k-1]*A + A+[2k-1]*A2

          = I+[2k-1]*A + A+[2k-1]*A       [Given, A2 = A]

          = I+ A+2[2k-1]*A

          = I+ A+[2k+1-2]*A

          = I+ [2k+1-1]*A

Thus, the result is true for n = k + 1

Hence, by principal of mathematical induction, the result is true for all natural numbers n.

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Wed April 19, 2017

Thankyou Mrs. Sneha Shidid .

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