Determinant 5x5

B<- matrix(c(2,0,0,0,0,3,1,0,1,0,0,1,1,2,0,2,1,2,1,1,1,2,3,0,2),5)
B

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    2    3    0    2    1
## [2,]    0    1    1    1    2
## [3,]    0    0    1    2    3
## [4,]    0    1    2    1    0
## [5,]    0    0    0    1    2

find the determinant by column 1

2* det(C)

where C is 4x4 matrix

C<- matrix(c(1,0,1,0,1,1,2,0,1,2,1,1,2,3,0,2),4)
C

##      [,1] [,2] [,3] [,4]
## [1,]    1    1    1    2
## [2,]    0    1    2    3
## [3,]    1    2    1    0
## [4,]    0    0    1    2

operate R3 -> R3 -R1

C reduces to

C<- matrix(c(1,0,0,0,1,1,1,0,1,2,0,1,2,3,-2,2),4)
C

##      [,1] [,2] [,3] [,4]
## [1,]    1    1    1    2
## [2,]    0    1    2    3
## [3,]    0    1    0   -2
## [4,]    0    0    1    2

find the determinant by C1

det(C) = 1* det(D)

where D is 3x3 matrix

D<- matrix(c(1,1,0,2,0,1,3,-2,2),3)
D

##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    1    0   -2
## [3,]    0    1    2

Apply the sarrus rule to matrix D and find the determinant

det(D) = 1*0 * 2 + 2 *-2* 0 + 3*1*1 -(2*1*2 + 1*-2*1 + 3* 0 *0) = 3 -2 = 1

det(B) = 2* det(C) = 2*1*det(D) = 2 * 1*1 = 2