Assignment 1

Define matrix

Question 01

A <- matrix (data=c(4,2,3,7,5,8),nrow=2,ncol=3,byrow=TRUE)
A

##      [,1] [,2] [,3]
## [1,]    4    2    3
## [2,]    7    5    8

B <-matrix (data=c(3,-2,4,6,9,-5),nrow=2,ncol=3,byrow=TRUE)
B

##      [,1] [,2] [,3]
## [1,]    3   -2    4
## [2,]    6    9   -5

1. Find A + B and A − B

A_plus_B <-A+B
A_plus_B

##      [,1] [,2] [,3]
## [1,]    7    0    7
## [2,]   13   14    3

A_minus_B <- A-B
A_minus_B

##      [,1] [,2] [,3]
## [1,]    1    4   -1
## [2,]    1   -4   13

2. A’A and AA’

t(A) %*% A

##      [,1] [,2] [,3]
## [1,]   65   43   68
## [2,]   43   29   46
## [3,]   68   46   73

A %*% t(A)

##      [,1] [,2]
## [1,]   29   62
## [2,]   62  138

Question 02 (a)Find (A + B)′ and A′ + B′ and compare them.

t(A+B)

##      [,1] [,2]
## [1,]    7   13
## [2,]    0   14
## [3,]    7    3

t(A)+t(B)

##      [,1] [,2]
## [1,]    7   13
## [2,]    0   14
## [3,]    7    3

Check Equality

all(t(A+B) ==(t(A) + t(B)))

## [1] TRUE

2. Show that (A′)′ = A

t(t(A))

##      [,1] [,2] [,3]
## [1,]    4    2    3
## [2,]    7    5    8

Check equality

all(t(t(A)) == A)

## [1] TRUE

Question 3 Define Matrix

A <- matrix(data=c(1,3,2,-1),nrow=2,ncol=2,byrow=TRUE)
A

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

B <- matrix(data=c(2,0,1,5),nrow=2,ncol=2,byrow=TRUE)
B

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

1. Find AB and BA

AB <- A %*% B
AB

##      [,1] [,2]
## [1,]    5   15
## [2,]    3   -5

BA <- B %*% A
BA

##      [,1] [,2]
## [1,]    2    6
## [2,]   11   -2

(b)Find |AB|, |A|, and |B|

det(A)

## [1] -7

det(B)

## [1] 10

det(AB)

## [1] -70

Verify that |AB| = |A||B|

det(A)*det(B)

## [1] -70

det(AB) == det(A)*det(B)

## [1] FALSE

Question 4 (a)Find A+B and tr(A+B)

A <- matrix(data=c(1,3,2,-1),nrow=2,ncol=2,byrow=TRUE)
A

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

B <- matrix(data=c(2,0,1,5),nrow=2,ncol=2,byrow=TRUE)
B

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

A_plus_B <- A+B
A+B

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

trace_A_plus_B <- sum(diag(A_plus_B))
trace_A_plus_B

## [1] 7

Question 5 Determine matrix

A <- matrix(data=c(1,2,3,2,-1,1),nrow=2,ncol=3,byrow=TRUE)
A

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

B <- matrix(data=c(3,-2,2,0,-1,1),nrow=3,ncol=2,byrow=TRUE)
B

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

(a)Find A+B and tr(A+B)

#A_plus_B <- A+B
#A_plus_B

#trace_A_plus_B <- sum(diag(A_plus_B))
#trace_A_plus_B

#print("A + B is not defined because the matrices have different dimensions")

(b)Compare tr(AB) and tr(BA)

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

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

BA <- B%*%A
BA

##      [,1] [,2] [,3]
## [1,]   -1    8    7
## [2,]    2    4    6
## [3,]    1   -3   -2

trace_AB <- sum(diag(AB))
trace_AB

## [1] 1

trace_BA <- sum(diag(BA))
trace_BA

## [1] 1

verify that tr(AB)=tr(BA))

sum(diag(AB))==sum(diag(BA))

## [1] TRUE