138 CHAPTER 8. MATRICES

8.4.1 The TransposeAnother 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 4

3 12 6

T

=

(1 3 24 1 6

)

What happened? The first column became the first row and the second column became thesecond row. Thus the 3×2 matrix became a 2×3 matrix. The number 3 was in the secondrow and the first column and it ended up in the first row and second column. Here is thedefinition.

Definition 8.4.4 Let A be an m× n matrix. Then AT denotes the n×m matrix which isdefined as follows. (

AT )i j = A ji

In words, the ith row becomes the ith column.

Example 8.4.5 (1 2 −63 5 4

)T

=

 1 32 5−6 4

 .

The transpose of a matrix has the following important properties.

Lemma 8.4.6 Let A be an m×n matrix and let B be a n× p matrix. Then

(AB)T = BT AT (8.14)

and if α and β are scalars,

(αA+βB)T = αAT +βBT (8.15)

Proof: From the definition,((AB)T

)i j

= (AB) ji

= ∑k

A jkBki

= ∑k

(BT )

ik

(AT )

k j

=(BT AT )

i j

The proof of Formula 8.15 is left as an exercise and this proves the lemma. ■

Definition 8.4.7 An n×n matrix, A is said to be symmetric if A = AT . It is said to be skewsymmetric if A =−AT .