5.1. MATRIX ARITHMETIC 91

= (a(AB)+b(AC))i j .

Thus A(B+C) = AB+AC as claimed. Formula 5.14 is entirely similar.Formula 5.15 is the associative law of multiplication. Using Definition 5.1.14,

(A(BC))i j = ∑k

Aik (BC)k j = ∑k

Aik ∑l

BklCl j

= ∑l(AB)il Cl j = ((AB)C)i j .

This proves 5.15. ■

5.1.5 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 5.1.22 Let A be an m× n matrix. Then AT denotes the n×m matrix which isdefined as follows. (

AT )i j = A ji

Example 5.1.23 (1 2 −63 5 4

)T

=

 1 32 5−6 4

 .

The transpose of a matrix has the following important properties.

Lemma 5.1.24 Let A be an m×n matrix and let B be a n× p matrix. Then

(AB)T = BT AT (5.16)

and if α and β are scalars,

(αA+βB)T = αAT +βBT (5.17)

Proof: From the definition,((AB)T

)i j= (AB) ji = ∑

kA jkBki = ∑

k

(BT )

ik

(AT )

k j =(BT AT )

i j

The proof of Formula 5.17 is left as an exercise. ■