## create a matrix
A <- matrix(c(3, 1, -1, 5, 2, 4,5,7,3,6,5,3), nrow=2, ncol=6)

## print the matrix
A

## get the (2,6)th element of the matrix
a.2.6 <- A[2,6]
a.2.6

## get row = 1 and colums 1,4,6
b <- A[1,c(1,4,6)]
b

## now subset the matrix A again and call it A
A1 <- A[, 1:3]
A1

## or subtract of the elements you dont want
A2 <- A[, -c(4:6)]
A2

## subtract A1 from A2
A3 <- A1 - A2
A3

## set A1 or A2 to A
A <- A1 

## multiply A by a scalar equal to 5
out <- 5*A
out 

## take the transpose of A
trans.A <- t(A)
trans.A

## create two more matrices
B <- matrix(c(-2,7,9), nrow=3, ncol=1)
C <- matrix(c(2,0,1,-1), nrow=2, ncol=2)

## matrix multiplication
AB <- A%*%B
AB
CA <- C%*%A
CA

## calculate the determinent of C
det.C <- det(C)
det.C

## the solve(A,b) funtion is used to solve Ax=b
## A is the matrix of coefficients and b can be a vector or a matrix
## if b is the identity matrix of the same size as A, then the solve command 
## gives the inverse of the matrix
inv.C <- solve(C)
inv.C

## the eigenvalue example
A <- matrix(c(1,1,0,3), nrow=2, ncol=2)
eig.A <- eigen(A)
eig.A

## get the eigenvalues
eig.A$values

## get the eigenvectors
eig.A$vectors

## check the length of the second eigenvector
e2 <- eig.A$vectors[,2]
l.e2 <- sqrt(t(e2)%*%e2)
l.e2

## now make a function for to calculate the trace of a matrix
trace <- function(X){
	out <- sum(diag(X))
	return(out)
}

trace(A)