C30 p150

For the matrix A, find A^2, A^3, A^4. Find the general formula for A^n for any positive integer.

So first, I need to make the matrix.

A <- matrix(c(1,0,2,1), nrow = 2, ncol = 2)
A
##      [,1] [,2]
## [1,]    1    2
## [2,]    0    1

Next, find A^2, A^3, A^4

A ** 2
##      [,1] [,2]
## [1,]    1    4
## [2,]    0    1
A ** 3
##      [,1] [,2]
## [1,]    1    8
## [2,]    0    1
A ** 4
##      [,1] [,2]
## [1,]    1   16
## [2,]    0    1

Since this matrix consists of 0 and 1 mostly, the only position with a general formula is the 2 which would be 2^n. So for any n in A^n, the matrix will be:

matrix(c(1,0,"2**n",1), nrow = 2, ncol = 2)
##      [,1] [,2]  
## [1,] "1"  "2**n"
## [2,] "0"  "1"