2.1. MATRICES 47

Another important operation on matrices is that of taking the transpose. The followingexample shows what is meant by this operation, denoted by placing a T as an exponent onthe matrix.  1 1 + 2i

3 1

2 6

T

=

(1 3 2

1 + 2i 1 6

)

What happened? The first column became the first row and the second column becamethe second row. Thus the 3 × 2 matrix became a 2 × 3 matrix. The number 3 was in thesecond row and the first column and it ended up in the first row and second column. Thismotivates the following definition of the transpose of a matrix.

Definition 2.1.16 Let A be an m × n matrix. Then AT denotes the n ×m matrix whichis defined as follows. (

AT)ij= Aji

The transpose of a matrix has the following important property.

Lemma 2.1.17 Let A be an m× n matrix and let B be a n× p matrix. Then

(AB)T= BTAT (2.16)

and if α and β are scalars,(αA+ βB)

T= αAT + βBT (2.17)

Proof: From the definition,((AB)

T)ij

= (AB)ji

=∑k

AjkBki

=∑k

(BT)ik

(AT)kj

=(BTAT

)ij

2.17 is left as an exercise. ■

Definition 2.1.18 An n × n matrix A is said to be symmetric if A = AT . It is said to beskew symmetric if AT = −A.

Example 2.1.19 Let

A =

 2 1 3

1 5 −3

3 −3 7

 .

Then A is symmetric.

Example 2.1.20 Let

A =

 0 1 3

−1 0 2

−3 −2 0

Then A is skew symmetric.