How can I fix this wrong result in matrix for loop in R?

You could try a slightly different approach here:

n <- 100
A <- matrix(0, n + 1, n + 1)

# create index for the diagonal and the "other" diagonales
A_diag <- matrix(c(1:(n + 1), 1:(n + 1)), ncol = 2)
A_diag_1 <- matrix(c(1:(n + 1), 1 + 1:(n + 1)), ncol = 2)
A_diag_minus_1 <- matrix(c(1:(n + 1), - 1 + 1:(n + 1)), ncol = 2)
A_diag_minus_2 <- matrix(c(1:(n + 1), - 2 + 1:(n + 1)), ncol = 2)

# populate those diagonales
A[A_diag] <- 1/2
A[A_diag_1[A_diag_1[,2] <= n + 1,]] <- 1/4
A[A_diag_minus_1[A_diag_minus_1[,2] >= 1,]] <- 1/6
A[A_diag_minus_2[A_diag_minus_2[,2] >= 1,]] <- 1/12

# create starting values
A[1, 1] <- 3/4
A[2, 1] <- 1/4

# calculate pA
pA <- A
for (i in 1:10000){ pA <- pA %*% A }

This returns

pA[1, 1:3]
#> [1] 0.2087122 0.1651514 0.1306823

For the calulation of pA you could use the expm package to speed things up:

library(expm)

pB <- A %^% 10000

pB[1,1:3]
#> [1] 0.2087122 0.1651514 0.1306823