Unitary Matrices: Solving system of Equation with Inverse Matrix

Compute the inverse of the coefficient matrix of the system of equations below and use the inverse to solve the system.

4x + 10y = 12 2x + 6y = 4

The inverse of a matrix plays the same roles in matrix algebra as the reciprocal of a number and division does in ordinary arithmetic:

we can solve a matrix equation like Ax=b for the vector x by multiplying both sides by the inverse of the matrix A

Ax=b ⇒ A−1Ax=A−1b ⇒ x=A−1b

The following system of equation illustrate the basic properties of the inverse of a matrix.

Load the matlib package

library(matlib)

Creating a 2 x 2 matrix

A <- matrix(c(4, 2, 10, 6), 2, 2)
b <- c(12,4)
showEqn(A, b)
## 4*x1 + 10*x2  =  12 
## 2*x1  + 6*x2  =   4

Finding Rank

A
##      [,1] [,2]
## [1,]    4   10
## [2,]    2    6
c( R(A), R(cbind(A,b)) )    
## [1] 2 2
# Finding the determinant of A
det(A)
## [1] 4

Finding the inverse of A

AInv <- inv(A)

AInv
##      [,1] [,2]
## [1,]  1.5 -2.5
## [2,] -0.5  1.0
# Proving A inverse X A is Identity matrix

 AInv %*% A
##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1

Identity Matrix * X is X

# I * X = AInv * b => X = AInv * b

X = AInv %*% b

X
##      [,1]
## [1,]    8
## [2,]   -2

Checking the answer

Solve(A, b, fractions = TRUE)
## x1    =   8 
##   x2  =  -2

Plot the equations

plotEqn(A,b)
## 4*x[1] + 10*x[2]  =  12 
## 2*x[1]  + 6*x[2]  =   4

### This intersection of two lines proves that this consistant system of equation has at least one solution.