Given:

A=

1 1 0

0 1 1

0 0 1

Now to compute A^10. Calculating rigorously all the powers of A down to 10 is time consuming and quite tedious, hence we will try to observe for a pattern in smaller powers of A such that in A^2, to derive A^n which is the general case and calculate or determine the case for n=10 in particular and we are done!

Let us compute A^2=

1 2 1

0 1 2

0 0 1

Then A^4=

1 4 6

0 1 4

0 0 1

Note that, in each of the above cases the diagonal entries are 1 and the (1,2)th and (2,3)th element is n in each case i.e., 2 and 4 respectively. Now the (1,3)th element is obtained by n(n-1)/2.

Hence A^n=

1 n n(n-1)/2

0 1 n

0 0 1

Therefore n=10, we have A^10=

1 10 45

0 1 10

0 0 1

Somjit Roy Answered question