Page 37 - Matrices theory and applications

2. Square Matrices
20
Proposition 2.2.1 Given M ∈ M n (A), the following assertions are
equivalent:
1. There exists N ∈ M n(A) such that MN = I n .
2. There exists N ∈ M n (A) such that N M = I n .

3. det M is invertible.
If M satisfies one of these equivalent conditions, then the matrices N, N
are unique and one has N = N .

Definition 2.2.1 One then says that M is invertible. One also says some-

times that M is nonsingular,or regular.One calls the matrix N = N the
inverse of M, and one denotes it by M −1 .If M is not invertible, one says
that M is singular.
Proof
Let us show that (1) is equivalent to (3). If MN = I n ,then det M ·
det N = 1; hence det M ∈ A .Conversely, if det M is invertible,
∗
(det M) −1 ˆ T is an inverse of M by (2.1). Analogously, (2) is equivalent
M
to (3). The three assertions are thus equivalent.
If MN = N M = I n , one has N =(N M)N = N (MN)= N .This

equality between the left and right inverses shows that these are unique.
Theset of theinvertible elementsof M n (A) is denoted by GL n (A)(for
“general linear group”). It is a multiplicative group, and one has
k −1
T −1
) .
(MN) −1 = N −1 M −1 , (M ) =(M −1 k (M ) =(M −1 T
) ,
T −1
The matrix (M ) is also written M −T .If k ∈ IN,one writes M −k =
k
k −1
j
(M ) and one has M M = M j+k for every j, k ∈ ZZ.
The set of the matrices of determinant one is a normal subgroup of
GL n (A), since it is the kernel of the homomorphism M → det M.Itis
called the special linear group and is denoted by SL n (A).
The orthogonal matrices are invertible, and they satisfy the relation
T
M −1 = M . In particular, orthogonality is equivalent to MM T = I n .
The set of orthogonal matrices with entries in a field K is obviously a
multiplicative group, and is denoted by O n (K). It is called the orthogonal
group. The determinant of an orthogonal matrix equals ±1, since
2
1= det M · det M T =(det M) .
The set SO n (K) of orthogonal matrices with determinant equal to 1 is
obviously a normal subgroup of the orthogonal group. It is called the special
orthogonal group. It is simply the intersection of O n(K)with SL n (K).
A triangular matrix is invertible if and only if its diagonal entries are
invertible; its inverse is then triangular of the same type, upper or lower.
The proposition below is an immediate application of Theorem 2.1.2.