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The **diagonal elements** at Row i, 1<=i<=100 are written down below in the table.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ...... | 100 |

1+3 | 0 | 2x3+1 | 4+3 | 0 | 2x6+1 | 7+3 | 0 | 100+3 |

Trace is the sum of the diagonal elements of a matrix so no need to bother about other elements. Moreover it's given that it is diagonal matrix so other elements=0.

**For all i%3==2 **the elements at d_{ii} will be 0. So they won't contribute anything to the trace.

**For all i%3==1 are i=1,4,7,10...100**

d_{ii}=i+3

The series will be :

(1+3)+(4+3) +(7+3)......(100+3)

How many such terms are there? 34 [from 1 to 99 there will be exactly 33 such terms whose mod3 will be 1 and then +1 for 100]

So, 34*3 + (1+4+7+10+....100)

Let a=1, d=3 then we can write this series as

34*3 + (a + (a+d) + (a+2d) + (a+3d) ..... + (a+33d) )

=102 + { 34*a + d(1+2+3+......33) }

=102 + {34 + 3*(33)(34)/2 } = 102 + 1717=1819.

**For all i%3==0, i's are 3,6,9....100**

d_{ii}=2i+1

There will be 33 such terms

So series is like :

(2*3+1) + (2*6+1)....(2*99+1)

=1*33 + 2(3+6+9....99)

=33 + 2*3(1+2+3...33)

=33 + 6*(33)*(34)/2 = 3399

**So total = 1819+3399=5218**

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we have to find** trace of matrix of order i , where 1<=i<=100.**

so from the question we can say that** ti follows a triplet** where.

if --->* imod3=0 : di =2i+1 [email protected]*

* imod3=1 : di=i+3 [email protected]*

* i mod3=2 : di=0 [email protected]*

di={7,13,19.................................199} [email protected]

di={4,7,10,....................................103} [email protected]

di = {0,0,0,0,0,...............................} [email protected]

therefore trace of matrix = **summation of di (1<=i<=100)**

=**using arithmetic progression sum eqn.**

= *(n/2) (first term +last term)*

**for eqn [email protected] : **

di={7,13,19.................................199} :** n=33, a=7 ,d=6, l=199**

**sum**= n/2 (a+l) = 33/2 (199+7) =**3399**

**for eqn @5 :**

di={4,7,10,....................................103} : **n=34 a=4 , d=3 ,l=103**

**sum **= n/2 (a+l) = 34/2 (4+103) =**1819**

**for eqn @6 :**

di = {0,0,0,0,0,...............................}

**sum =0**