Monday, March 10 − Lecture 25 : Invertible matrices.
1. Define the inverse of a matrix.
2. Find the inverse of a matrix.
3. Solve a system by using the inverse of a matrix.
4. Recognize that a matrix is invertible if and only if its inverse has the RREF.
The multiplicative matrix inverse A of A.
25.1 Definition − We say that an n × n matrix A is invertible (or non-singular) if there
exists an n × n matrix B such that both conditions AB = I =nBA hold true. We call such a
matrix B, an inverse of A.
Note that only a pair of square matrices of equal dimension can satisfy the condition AB
= BA. So if we refer to matrix A as being invertible then A must be a square matrix
25.1.1 Definition – If A is an n × m matrix for which there exists an m × n matrix B
such that AB = I n then we say that B is a right inverse of A and A is a left inverse of
B. If B is both a right inverse and left inverse of A then, by definition of “inverse”,
both A and B are easily seen to be both (square) invertible matrices.
25.1.2 Proposition − Uniqueness of the inverse of a matrix. Let A be an n × n
invertible matrix. Then there exists no more than one inverse B of A.
Proof : Suppose A is invertible. Suppose both B and C are inverses of the matrix A.
Then AB = I = BA and AC = I = CA.
C = CI = C(AB) = (CA)B = IB = B.
So C = B.
Then if a matrix A is invertible it will have, at most, one inverse.
25.1.3 Notation − For an invertible matrix A, the unique inverse matrix of A is denoted
by A -1.
If a matrix has dimensions 3 × 3 or larger it is not easy to tell whether it has an inverse
or not simply by looking at its entries. Even if we know a matrix is invertible finding
its unique inverse can still be a considerable amount of work. However, if a matrix is
2 × 2 it is fairly straightforward as the following example shows. 25.1.4 Example − Show that if A is the matrix,
such that ad − bc is not zero, A exists and is equal to the matrix :
Solution: We compute the product AB:
Then AB = I. Computation BA = I follows similar steps. This means that if ad − bc ≠ 0
then a two by two matrices has an inverse and is of the form given to the matrix B.
The converse : If ad – bc = 0 then A cannot be invertible. To see this, suppose both ad
– bc = 0 and A exists. Then
Then for any matrix B, AB will produce a two by two matrix whose second row only
contains zeros and so cannot be the identity matrix.
25.1.5 Definition − The number ad − bc obtained in the 2 × 2 matrix A above is
called the determinant of the 2 × 2 matrix A. It is denoted by det A.
- We can always use the derived expression as a way of finding the inverse of a 2 by
2 provided det A ≠ 0.
The inverse of a matrix A, when it exists, can be quite useful when trying to solve systems
of linear equations Ax = b. This is shown in the following proposition.
25.2 Theorem − If A is invertible and Ax = b is consistent then the only solution to the
system Ax = b is x = A b.1
Proof : Given: A is an n × n invertible matrix and Ax = b is consistent. If the vector uis a solution to Ax = b and A is invertible then
u = Inu = (A A)u = A (Au) = A b. −1
So u = A b is the only solution of Ax = b.
25.2.1 Corollary – If A is an invertible n × n matrix then A RREF = I nnd so Rank(A) = n.
Proof : Given: A is an n × n invertible matrix.
Since Ax = 0 is consistent and A is invertible, by the theorem, the system Ax = 0 has a
unique solution u = A 0. Since the solution to Ax = 0 is unique, the matrix A RREF
points to no free variables. This means A has a leading-1 in each of its n columns
and each of its n rows. Hence A RREF = In. By definition, Rank(A) = n, as required.
25.2.2 Corollary – If A and B are an n × n matrices and AB = I then Anis invertible.
Proof : Given : A and B are an n × n matrices and AB = I n Required to show: BA = I . n
If B is as defined above, we construct the homogeneous system Bx = 0. Let u be any