## 1.6 Taking the inverse of a square matrix

The inverse of a matrix is denoted $$\mathbf{A}^{-1}$$. You can think of the inverse of a matrix like $$1/a$$. $$1/a \times a = 1$$. $$\mathbf{A}^{-1}\mathbf{A} = \mathbf{A}\mathbf{A}^{-1} = \mathbf{I}$$. The inverse of a matrix does not always exist; for one it has to be square. We’ll be using inverses for variance-covariance matrices and by definition (of a variance-covariance matrix), the inverse of those exist. In R, there are a couple way common ways to take the inverse of a variance-covariance matrix (or something with the same properties). solve() is the most common probably:

A = diag(3, 3) + matrix(1, 3, 3)
invA = solve(A)
invA %*% A
             [,1]          [,2] [,3]
[1,] 1.000000e+00 -6.938894e-18    0
[2,] 2.081668e-17  1.000000e+00    0
[3,] 0.000000e+00  0.000000e+00    1
A %*% invA
             [,1]          [,2] [,3]
[1,] 1.000000e+00 -6.938894e-18    0
[2,] 2.081668e-17  1.000000e+00    0
[3,] 0.000000e+00  0.000000e+00    1

Another option is to use chol2inv() which uses a Cholesky decomposition:

A = diag(3, 3) + matrix(1, 3, 3)
invA = chol2inv(chol(A))
invA %*% A
              [,1]         [,2]          [,3]
[1,]  1.000000e+00 6.938894e-17  0.000000e+00
[2,]  2.081668e-17 1.000000e+00 -2.775558e-17
[3,] -5.551115e-17 0.000000e+00  1.000000e+00
A %*% invA
             [,1]          [,2]          [,3]
[1,] 1.000000e+00  2.081668e-17 -5.551115e-17
[2,] 6.938894e-17  1.000000e+00  0.000000e+00
[3,] 0.000000e+00 -2.775558e-17  1.000000e+00

For the purpose of this course, solve() is fine.