7.2. Linear Equations

Solving a linear equation:

> A <- matrix(1:4, nrow=2)
> v <- c(1:2)
> b <- A %*% v
> A
     [,1] [,2]
[1,]    1    3
[2,]    2    4
> v
[1] 1 2
> b
     [,1]
[1,]    7
[2,]   10
> solve(A, b)
     [,1]
[1,]    1
[2,]    2

Computing the inverse of a matrix through solving the equation \(AX=I\):

> A <- matrix(1:4, nrow=2)
> solve(A)
     [,1] [,2]
[1,]   -2  1.5
[2,]    1 -0.5

Computing the quadratic form \(x^T A^{-1} x\):

> A <- matrix(1:4, nrow=2)
> x <- c(2,3)
> x %*% solve(A) %*% x
     [,1]
[1,]  2.5
> x %*% solve(A, x)
     [,1]
[1,]  2.5

The second approach is much more efficient and reliable.