88 CHAPTER 5. MATRICES

The second entry of this m×1 matrix is

A21B1 j +A22B2 j + · · ·+A2nBn j =m

∑k=1

A2kBk j.

Similarly, the ith entry of this m×1 matrix is

Ai1B1 j +Ai2B2 j + · · ·+AinBn j =m

∑k=1

AikBk j.

This shows the following definition for matrix multiplication in terms of the i jth entries ofthe product coincides with Definition 5.1.11.

Definition 5.1.14 Let A = (Ai j) be an m×n matrix and let B = (Bi j) be an n× p matrix.Then AB is an m× p matrix and

(AB)i j =n

∑k=1

AikBk j ≡ Ai1B1 j +Ai2B2 j + · · ·+AinBn j. (5.12)

Another way to write this is

(AB)i j =

1×n(Ai1 Ai2 · · · Ain

)n×1B1 j

B2 j...

Bn j

Note that to get (AB)i j you multiply the ith row of A and the jth column of B. In terms ofthe dot product from calculus or earlier in this book, the i jth entry of AB is the dot productof the ith row of A with the jth column of B.

I will summarize the above discussion in the following proposition which shows that theabove definition delivers the earlier one about AB =

(Ab1 · · · Abp

). It is important

to realize these two definitions are equivalent.

Proposition 5.1.15 Let A be an m×n matrix. Let B =(

b1 · · · bp

)where each bk is

a column vector or n×1 matrix so B is an n× p matrix. Then AB is an m× p matrix and

AB =(

Ab1 · · · Abp

)so the kth column of AB is just Abk.

Proof: From the definition of multiplication of matrices, (AB)ik = ∑r AirBrk. However,

bk =

B1k

...Bnk

