Matrix Inversion! theorems

Theorem 5:
If A is an invertible n × n matrix, then for each b in R, the equation Ax = b has the unique solution x = Ab.

Proof

To show that the solution does exist, we replace x in Ax = b with Ab so

Ax = A(Ab) = (AA)b = Ib = b

To show that there are no other solutions we call a solution u

So:
Au = b
multiply both sides by A:
A Au = Ab
because AA = I :
Iu = Ab
and finally:
u = Ab

Therefore we know that Ab is a solution of x in equation Ax = b and further that it is the only solution of x proving theorem 5.

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Theorem 6:

a.If A is an invertible matrix then A is invertible and

(A) = A

b. If A and B are n × n invertible matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order.

(AB) = BA

c. If A is an invertible matrix, then so is A, and the inverse of A is the transpose of A.

(A) = (A)

Proof

The first statement probably seems sensible to you. If we replace A with any number the identity works just as well. Let's replace A with 2 then we get

(2) = (½) = 2 or (2) =2* = 2

However, since A is a matrix, not a real number, we cannot prove it in the same way. Instead we must find a matrix C so that AC = I and also CA= I

Because we know that A A= I , then we know that C must be A so A is invertible and its inverse is A.

For the second statement we want to show that AB and BA are in fact inverses or equivalently that their product equals the identity matrix

(AB)(BA) = A(BB)A = AIA = AA = I and (BA)(AB) = I

Let's use Theorem 3(d) from Chapter 2 of this book, which says (AB) = BA

(A)A = (AA) = I = I and A(A)= I

so A is invertible with an inverse of (A)

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Theorem 7:

An n × n matrix A is invertible if and only if A is row equivalent to I, and any sequence of elementary row operations that reduces A to I also transforms I into A.

The theorem is stating that if any row operations (switching rows, multiplying by a constant, or adding/subtracting a multiple of another row) taking some matrix to the identity exist, those very operations can bring the identity into the inverse of that matrix we started out with.

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Theorem 8:

The Invertible Matrix Theorem

Let A be a square n × n matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false.

  1. A is an invertible matrix.
  2. A is row equivalent to the n × n identity matrix.
  3. A has n pivot positions.
  4. The equation Ax = 0 has only the trivial solution.
  5. The columns of A form a linearly independent set.
  6. The linear transformation x -> Ax is one-to-one.
  7. The equation Ax = b has at least one solution for each b in R.
  8. The columns of A span R.
  9. The linear transformation x -> Ax maps R onto R.
  10. There is an n × n matrix C such that CA = I.
  11. There is an n × n matrix D such that AD = I.
  12. A is an invertible matrix.

The truth of any of these statements implies the truth of the remainder of them. Each and every one of these twelve statements imply each other. It's pretty remarkable that knowing just one thing about a matrix can tell you so much about it.

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