# Entrance exams

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Find A^10 where A is the matrix

1   1   0

0   1   1

0   0   1

0

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