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p = (1, 2)T

Matrix Inverse

It was tedious to figure that out (and would be much worse if A were 5 × 5). It would be nice to have a better way. Say that

q = Ap               (1)

for column vectors p and q and N×N matrix A . We know A and q. We are trying to figure out what p must be.

What if there were a matrix BN×N such that

p = Bq               (2)

If there were such a matrix, then we could calculate what we want ( p ) from q.

Substitute (2) into (1):

q = A(Bq)           (3)
q = (AB)q           (4)

If (4) is true, then (AB) = I. (Remember that I is unique). B, if it exists, is the inverse of A, written A-1.

So now you can solve

q = Ap               (1)

for p by multiplying each side by A-1:

A-1q = A-1Ap
A-1q = p

Of course, the above assumes that you somehow managed to calculate A-1.


For any square matrix A, is there always going to be an inverse A-1 ? Hint: consider the zero matrix.