CUNY Math Bridge - Wk 1, Linear Algebra only

Linear Algebra

1. Using matrix operations, describe the solutions for the following family of equations:

x + 2y - 3z = 5

2x + y - 3z = 13

-x + y + 2z= -8

1. Find the inverse of the above 3x3 (non-augmented) matrix.

A=matrix(c(1,2,-1,2,1,1,-3,-3,2),3,3)
print(A)

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

Inverse.matrix = solve(A)
print(Inverse.matrix)

##            [,1]      [,2] [,3]
## [1,] -0.8333333 1.1666667  0.5
## [2,]  0.1666667 0.1666667  0.5
## [3,] -0.5000000 0.5000000  0.5

2. Solve for the solution using R.

A=matrix(c(1,2,-1,2,1,1,-3,-3,2),3,3)
B= c(5,13,-8)
x <- solve(A)%*%B
print(x)

##      [,1]
## [1,]    7
## [2,]   -1
## [3,]    0

3. Modify the 3x3 matrix such that there exists only one non-zero variable in the solution set.

v_A <- c(1,2,1,2,1,1,5,13,-8)
A <- matrix(v_A, 3,3)
v_B <- c(5,13,-8)
B <- matrix(v_B, 3,1)
A1 <- solve(A)
X <- round(A1 %*% B,0)
X

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

2. Consider the matrix, q=matrix(c(3,1,4,4,3,3,2,3,2),nrow=3). Let b=c(1,4,5). Use Cramer’s rule and R to determine the solution, x, to qx=b, if one exists. Show all determinants.

q<-matrix(c(3,1,4,4,3,3,2,3,2), nrow=3)
print(q)

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

detrm.q<-det(q)
print(paste("Dtrm q = ", detrm.q))

## [1] "Dtrm q =  13"

qx<-matrix(c(1,4,5,4,3,3,2,3,2), nrow=3)
detrm.x<-det(qx)
print(paste("Dtrm x = ", detrm.x))

## [1] "Dtrm x =  19"

qy<-matrix(c(3,1,4,1,4,5,2,3,2), nrow=3)
detrm.y<-det(qy)
print(paste("Dtrm y = ", detrm.y))

## [1] "Dtrm y =  -33"

qz<-matrix(c(3,1,4,4,3,3,1,4,5), nrow=3)
detrm.z<-det(qz)
print(paste("Dtrm z = ", detrm.z))

## [1] "Dtrm z =  44"

print(paste("The determinants are ", detrm.q, ", " , detrm.x,",", detrm.y, ", and ",detrm.z))

## [1] "The determinants are  13 ,  19 , -33 , and  44"

Solution using r

q=matrix(c(3,1,4,4,3,3,2,3,2),nrow=3)
b=c(1,4,5)
x <- solve(q)%*%b
print(x)

##           [,1]
## [1,]  1.461538
## [2,] -2.538462
## [3,]  3.384615