M is a matrix
Let m =
[
1 1 2
0 1 1
0 0 1
]
Find M^n where n is a positive integer.
Hello, Mr_Green!
I used a very primitive method . . .
M is a matrix
. - . - . . | 1 1 2 |
. . M .= .| 0 1 1 |
. - . - . . | 0 0 1 |
Find M^n where n is a positive integer.
I cranked out the first powers and looked for a pattern.
. . . . . . . . | 1 2 5 |
. . M^2 .= .| 0 1 2 |
. . . . . . . . | 0 0 1 |
. . . . . . . . | 1 3 9 |
. . M^3 .= .| 0 1 3 |
. . . . . . . . | 0 0 1 |
. . . . . . . . | 1 4 14 |
. . M^4 .= .| 0 1 -4o|
. . . . . . . . | 0 0 -1o|
. . . . . . . . | 1 5 20 |
. . M^5 .= .| 0 1 -5o|
. . . . . . . . | 0 0 -1o|
. . . . . . . . . . . . | 1 . n . n(n+3)/2 | .**
Hence: -M^n -= -| 0 . 1 . . . - n - - |
. . . . . . . . . . . . | 0 . 0 . . . - 1 . . |
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
**
I worked out the formula for the a13 element
. . by examining the sequence: 2, 5, 9, 14, 20, ...
. . and constructing a quadratic function for it.