80 CHAPTER 4. MATRICES

Proposition 4.1.2 If all multiplications and additions make sense, the following hold formatrices, A,B,C and a,b scalars.

A(aB+bC) = a(AB)+b(AC) (4.9)

(B+C)A = BA+CA (4.10)

A(BC) = (AB)C (4.11)

Proof: Using the above definition of matrix multiplication,

(A(aB+bC))i j = ∑k

Aik (aB+bC)k j = ∑k

Aik(aBk j +bCk j

)= a∑

kAikBk j +b∑

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

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

showing that A(B+C) = AB+AC as claimed. Formula 4.10 is entirely similar.Consider 4.11, the associative law of multiplication. Before reading this, review the

definition of matrix multiplication in terms of entries of the matrices.

(A(BC))i j = ∑k

Aik (BC)k j = ∑k

Aik ∑l

BklCl j

= ∑l(AB)il Cl j = ((AB)C)i j .■

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 12 6

T

=

(1 3 2

1+2i 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. This motivatesthe following definition of the transpose of a matrix.

Definition 4.1.3 Let A be an m× n matrix. Then AT denotes the n×m matrix which isdefined as follows. (

AT )i j = A ji

The transpose of a matrix has the following important property.

Lemma 4.1.4 Let A be an m×n matrix and let B be a n× p matrix. Then

(AB)T = BT AT (4.12)

and if α and β are scalars,

(αA+βB)T = αAT +βBT (4.13)