Multivariate Analysis

Matrix Inversion Example

Inversion by Hand

First we will augment Matrix A with an identity matrix and call the result A₀.

We will also need an identity matrix called I₀.

Step 1

Now we will work on the first column of I₀. The diagonal element in the first column of A₀ is called the pivot. We will replace the first diagonal element of I₀ with 1/a₁₁. Then we will replace the non-pivot elements with -a_ij/a₁₁ and call the result I₁.
Computer A₁ = I₁ * A₀.

Step 2

Now we repeat the process by modifying the second column of I₀. This time the pivot value is the 22 element. The new matrix is called I₂.
Compute A₂ = I₂ * A₁.

Step 3

Lastly we repeat the process by modifying the third column of I₀. This time the pivot value is the 33 element. The new matrix is called I₃.
Compute A₃ = I₃ * A₂.

The last three columns of A₃ are the inverse of A, thus:

Stata Example

matrix A = (4,2,2 \ 4,6,8 \ -2,2,4)

matrix list A

A[3,3]
    c1  c2  c3
r1   4   2   2
r2   4   6   8
r3  -2   2   4

matrix A1 = inv(A)

matrix list A1

A1[3,3]
      r1    r2    r3
c1     1   -.5    .5
c2    -4   2.5    -3
c3   2.5  -1.5     2

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