Finding inverse matrix in R

Question

I have a variance covariance matrix S:

> S
     [,1] [,2]
[1,]    4   -3
[2,]   -3    9

I am trying to find an inverse of it.

The code I have is:

>invS <- (1/((S[1,1]*S[2,2])-(S[1,2]*S[2,1])))*S
           [,1]       [,2]
[1,]  0.1481481 -0.1111111
[2,] -0.1111111  0.3333333

However, if I use solve(), I get this:

>invSalt <- solve(S)
          [,1]      [,2]
[1,] 0.3333333 0.1111111
[2,] 0.1111111 0.1481481

Why is invS incorrect? What should I change to correct it?

Answer 1

You correctly found the determinant in the denominator, but the rest is wrong.

Off-diagonal elements should be with the opposite sign, while the diagonal elements should be switched. Both of those things are clearly visible when comparing the two matrices.

That's not the most convenient thing to do by hand, so solve is really better. If you insist on doing it manually, then you could use

matrix(rev(S), 2, 2) / (prod(diag(S)) - S[1, 2] * S[2, 1]) * (2 * diag(1, 2) - 1)
#           [,1]      [,2]
# [1,] 0.3333333 0.1111111
# [2,] 0.1111111 0.1481481

Answer 2

The correct formula is

 (1/((S[1,1]*S[2,2])-(S[1,2]*S[2,1])))* matrix(c(S[2,2], -S[2,1], -S[1,2], S[1,1]),2)