THE INVERSE OF A MATRIX
8.3
THE INVERSE OF A MATRIX
(LIKE “DIVISION”)
If A is n x n matrix and let In be the n x n identity
matrix. If there exists a matrix A-1 such that
AA-1 = In = A-1A
Then A-1 is called the inverse of A.
Note: A-1 is read “A inverse”. -1 is NOT an exponent
here!
Note: Not every matrix has an inverse. In order to
have an inverse a matrix must be a square
matrix. But still not every square matrix has an
inverse.
TO FIND THE INVERSE OF A MATRIX
  Augment
  Perform
  The
A with I.
row operations until left side looks like I.
right side is A-1.
Note: There is a fairly simple formula to find the
inverse of a 2x2 matrix if it exists.
A UNIQUE SOLUTION FOR A SYSTEM OF
EQUATIONS…
If A is an invertible matrix (it has an inverse), then
the system of linear equations represented by
AX = B has a unique solution given by
X = A-1B.
Proof:
AX = B
A-1AX = A-1B
IX = A-1B
X = A-1B