Solving Systems of Linear Equations – excersize C34

C34 Find all solutions to the linear system: #x + y ??? z = ???5 #x ??? y ??? z = ???3 #x + y ??? z = 0

To find all the soultions ginv() function Calculates the Moore-Penrose generalized inverse of a matrix X

library(matlib) #use the package
## Warning: package 'matlib' was built under R version 3.4.3
 A <- matrix(c(1,1,1,1,-1,1,-1,-1,-1),nrow = 3,byrow = TRUE)
 B <- c(-5,-3,0) 
 showEqn(A, B)
##  1*x1 + 1*x2 + 1*x3  =  -5 
##  1*x1 - 1*x2 + 1*x3  =  -3 
## -1*x1 - 1*x2 - 1*x3  =   0
 library(MASS)
## Warning: package 'MASS' was built under R version 3.4.3
x <- ginv(A) %*% B
x
##        [,1]
## [1,] -1.375
## [2,]  0.250
## [3,] -1.375

This gives the vaue of x = -1.375 , y = 0.250 , z = -1.375

To check if the systems of equation is consistent or not we can use the below procedure

c(R(A), R(cbind(A,B)))   # show ranks
## [1] 2 3
all.equal( R(A), R(cbind(A,B)) ) # show consistency
## [1] "Mean relative difference: 0.5"
plotEqn3d(A,B)

The ranks here are not same and consitency shows difference is 0.5 so this system of equations is not consistent.