2.3 - Matrix Inverses
For (nn) matrices A, B, if
AB = BA = I
then B = A–1 is the inverse matrix of A.
A matrix that possesses an inverse is invertible. A non-invertible matrix is singular.
Example 2.3.1
Show that  is the inverse of  .
and
Therefore B is the inverse matrix of A .
If the inverse to a matrix A exists, then it is unique.
Proof:
Suppose that matrices B and C are both inverses of A . Then
AB = BA = I and AC = CA = I.
C = IC = (BA)C = B(AC) = BI = B
The inverse matrix, if it exists, is therefore unique.
From Example 2.2.6 above,  is its own inverse for all values of the real number k:  .
Therefore in this case A–1 = A , even though A is not ± I .
The uniqueness of the inverse allows us to check just one of A–1A = I or AA–1 = I .
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