Why matrices have only one inverse

inverse_matrix_2

If we define the inverse of a matrix A as another matrix A−1 such that

A−1.A=I

then it is true, but not obvious, that also

A.A−1=I.

Remember: I is the identity matrix, the matrix which corresponds to the transformation which is just leaving things exactly as they are.

identity2

If A−1.A=I, then multiply each side before by A and after by A−1.

A.A−1.A.A−1 = A.A−1

If A−1 exists, then det A ≠ 0, so det A−1 ≠ 0 too, and det A.A−1 ≠ 0. So A.A−1 has an inverse. Call it B. Then:

B.A.A−1.A.A−1 = B.A.A−1

Which means

A.A−1 = I     ▇

In other words, the general truth that with matrices M.N may not be equal to N.M doesn’t apply with inverses. A.A−1 is the same as A−1.A. Just as well, because otherwise you would have to calculate two inverses for each matrix, one for multiplying before, and another for multiplying after.

Another fact about inverses:

If A and B have inverses, then:

(AB)−1 = B−1.A−1

You can see why by thinking about socks and shoes.

The inverse of [put on shoes].[put on socks].[feet]

is [take off socks].[take off shoes].[feet].