Inverse Matrix Theorem

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Inverse matrix theorem

Brian Krummel September 13, 2019

Theorem 1 (Inverse matrix theorem). Let A be an n × n matrix. The following statements are logically equivalent:

(a) A is an invertible matrix.

(b) A is row equivalent to the n × n identity matrix.

(d) The equation Ax = 0 has only the trivial solution.

(e) The columns of A are linearly independent.

(f) The linear transformation T (x) = Ax is one-to-one.

(g) The equation Ax = b has at least one solution for each b in Rn.

(h) The columns of A span Rn. (i) The linear transformation T (x) = Ax is onto.

(j) There is an n × n matrix C such that CA = I.

(k) There is an n × n matrix D such that AD = I.

(`) AT is an invertible matrix.

Let us look at this theorem carefully:

• (a) ⇔ (b) by a theorem from last lecture, the reduced echelon form of an invertible matrix. In particular, we showed that if A is invertible then Ax = b has a unique solution for each b in Rn. It followed that A must be a square matrix with a pivot in every row and in every column, that is the reduced echelon form of A is the identity matrix. On the other hand, using elementary matrices we showed that if the reduced echelon form of A is the identity matrix, then A is invertible.

• (b) ⇔ (c): Clearly if the reduced echelon form of an n × n matrix A is the identity matrix, then A has n pivot position. In fact, the only n × n reduced echelon form with n pivots is the identity matrix. Now (a)–(c) are all equivalent.

1 • From Section 1, we know that (d), (e), (f), and (c) are all equivalent statements for uniqueness “Ax = 0 has only the trivial solution.” Here we restate (c) as “A has n pivot positions in each column.”

• Also from Section 1, we know that (g), (h), (i), and (c) are all equivalent statements for existence “Ax = b has at least one solution for each b in Rn.” Here we restate (c) as “A has n pivot positions in each row.” Now (a)–(i) are all equivalent.

• Notice that the theorem is stating that existence of solutions to Ax = b is logically equivalent to uniqueness of solutions to Ax = b. This is a very special property of square n×n matrices and does not hold true for non-square matrices.

• By the deﬁnition of inverse matrix, (a) implies both (j) and (k). The theorem asserts that knowing either (j) or (k) implies (a).

• (j) ⇒ (d): If there is a matrix C such that CA = I, then we can multiply both sides of Ax = 0 by C to obtain x = Ix = CAx = C0 = 0 and thus the only solution is the trivial solution x = 0. Since we showed above that (a) ⇔ (d), it follows from (j) ⇒ (d) that (j) ⇒ (a).

• (k) ⇒ (g): If there is a matrix D such that AD = I, then x = Db is a solution to Ax = b for each b in Rm: Ax = ADb = Ib = b. Since we showed above that (a) ⇔ (g), it follows from (k) ⇒ (g) that (k) ⇒ (a). Now we know that (a)–(k) are all equivalent.

• (a) ⇔ (`): By the properties of invertible matrices we know that if A is invertible, then AT is also invertible. But A = (AT )T so by the same reasoning if AT is invertible, then A = (AT )T is also invertible.