- #1
U.Renko
- 57
- 1
Homework Statement
find a formula for [itex] \begin{bmatrix}
1 & 1& 1\\
0& 1& 1\\
0& 0 & 1
\end{bmatrix} ^n[/itex]
and prove it by induction
the induction part is ok.
I'm just having trouble finding a pattern
I may have figured it out but it looks too cumbersome
Homework Equations
The Attempt at a Solution
Lets call that matrix A
I computed A^2 through A^5 and noticed a pattern:
[itex] A^2 = \begin{bmatrix}
1 & 2&3\\
0& 1& 2\\
0& 0 & 1
\end{bmatrix} [/itex]
[itex] A^3 = \begin{bmatrix}
1 & 3& 6\\
0& 1& 3\\
0& 0 & 1
\end{bmatrix} [/itex]
[itex] a^4 = \begin{bmatrix}
1 & 4& 10\\
0& 1& 4\\
0& 0 & 1
\end{bmatrix} [/itex]
so the pattern is :
below the diagonal is always 0
the diagonal is always 1
[itex] a_12 = a_23 = n [/itex]
[itex] a_13 = some number [/itex] that's where I had trouble figuring the pattern
I noticed that, it is also the sum of the elements in the first row of A^(n-1) but that is a bit awkward to generalize.
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