Linear Algebra, Theory And Applications, 2012a

= ∑ k
(
B
T ) ik
(
A
T ) kj
2.1. MATRICES 47
Proof: From the definition,
(
(AB) T ) ij
= (AB) ji
= ∑ k
A jk B ki
= ( B T A T ) ij
(2.17) is left as an exercise.
Definition 2.1.18 An n × n matrix A is said to be symmetric if A = A T . It is said to be
skew symmetric if A T = −A.
Example 2.1.19 Let
Then A is symmetric.
Example 2.1.20 Let
Then A is skew symmetric.
⎛
A = ⎝ 2 1 3 ⎞
1 5 −3 ⎠ .
3 −3 7
⎛
A = ⎝
0 1 3
−1 0 2
−3 −2 0
⎞
⎠
There is a special matrix called I anddefinedby
I ij = δ ij
where δ ij is the Kronecker symbol defined by
{ 1ifi = j
δ ij =
0ifi ≠ j
It is called the identity matrix because it is a multiplicative identity in the following sense.
Lemma 2.1.21 Suppose A is an m × n matrix and I n is the n × n identity matrix. Then
AI n = A. If I m is the m × m identity matrix, it also follows that I m A = A.
Proof:
(AI n ) ij
= ∑ k
A ik δ kj
= A ij
and so AI n = A. The other case is left as an exercise for you.
Definition 2.1.22 An n × n matrix A has an inverse A −1 if and only if there exists a
matrix, denoted as A −1 such that AA −1 = A −1 A = I where I =(δ ij ) for
{ 1 if i = j
δ ij ≡
0 if i ≠ j
Such a matrix is called invertible.