Find the rank and nullity of the matrix

\[ A = \begin{bmatrix} 3 & 2 & 1 & 1 & 1 \\ 2 & 3 & 0 & 1 & 1 \\ -1 & 1 & 2 & 1 & 0 \\ 1 & 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 2 & -1 \end{bmatrix} \]

library(pracma)
## Warning: package 'pracma' was built under R version 4.2.3
A <- matrix(c(3, 2, 1, 1, 1, 2, 3, 0, 1, 1, -1, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 2, -1), nrow = 5, byrow = TRUE)
A
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    3    2    1    1    1
## [2,]    2    3    0    1    1
## [3,]   -1    1    2    1    0
## [4,]    1    1    0    1    1
## [5,]    0    1    1    2   -1

With n = 5 columns and r = 5 nonzero rows. The rank is r(A) = 5 The nullity is n(E) = 5 - 5 = 0

rref(A)
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    1    0    0    0    0
## [2,]    0    1    0    0    0
## [3,]    0    0    1    0    0
## [4,]    0    0    0    1    0
## [5,]    0    0    0    0    1