1. Problem set 1

Solution 1

library(matlib)
## Warning: package 'matlib' was built under R version 3.5.1

Create the equation:

A <- matrix(c(1, 3, 5,
             1, 2,4,
             1,0,1), 3, 3, byrow=TRUE)

A
##      [,1] [,2] [,3]
## [1,]    1    3    5
## [2,]    1    2    4
## [3,]    1    0    1
A_T <- t(A)
A_T
##      [,1] [,2] [,3]
## [1,]    1    1    1
## [2,]    3    2    0
## [3,]    5    4    1
A1 <- t(A)%*%A
A1
##      [,1] [,2] [,3]
## [1,]    3    5   10
## [2,]    5   13   23
## [3,]   10   23   42
A2 <- A%*%t(A)
A2
##      [,1] [,2] [,3]
## [1,]   35   27    6
## [2,]   27   21    5
## [3,]    6    5    2

Solution 2:

A <- matrix(c(1, 0, 0,
             0, 1,0,
             0,0,1), 3, 3, byrow=TRUE)

A
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1
A_T <- t(A)
A_T
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1
A1 <- t(A)%*%A
A1
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1
A2 <- A%*%t(A)
A2
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1