A First Course in Linear Algebra, Beezer, R., 2008
Question C23 page 278:
Doing the computations by hand, find the determinant of the matrix below.

1 3 2
4 1 3
1 0 1

Define matrix
Below a 3x3 matrix is coded and printed.

A <- matrix(c(1, 4, 1, 3, 1,0 ,2 ,3 ,1 ), nrow = 3, byrow = FALSE)
print(A)
##      [,1] [,2] [,3]
## [1,]    1    3    2
## [2,]    4    1    3
## [3,]    1    0    1

Code the coordinates
Below the coordinates are coded to prepare for determinant calculation.

a11<-A[1,1]
a12<-A[1,2]
a13<-A[1,3]
a21<-A[2,1]
a22<-A[2,2]
a23<-A[2,3]
a31<-A[3,1]
a32<-A[3,2]
a33<-A[3,3]

Determinant calculation
This is the determinant calculation.

#Calculation
D_a<-  (a11*a22*a33)  +
       (a21*a32*a13)  +
       (a31*a12*a23)  - 
       (a13*a22*a31)  - 
       (a23*a32*a11)  -
       (a31*a12*a21)
print(D_a)
## [1] -4

Proof
I validated the above calculation was correct by using the det() function.
Both the manual calucation and the function equal -4

D_a_validate <- det(A)

D_a_validate
## [1] -4