Math 300
Notes for Section 2.1
1. (Matrices). An mnmatrix Ais a rectangular array of numbers arranged in mhorizontal rows
and nvertical columns:
AD
2
6
6
6
4
a11 a12 ::: a1n
a21 a22 ::: a2n
:
:
::
:
::
:
:
am1 am2 ::: amn
3
7
7
7
5
:
We say thatAis of size mby n. The number aij in the ith row and jth column of Ais called the
.i;j/entry of Aand we write
ADŒaij :
If mDn, we say that Aisasquare matrix of order n.
2. (Equality of Matrices). Two matrices are said to be equal if they have the same size and their
corresponding entries are equal.
3. (Matrix Addition and Subtraction). If ADŒaij and BDŒbij are matrices of the same size,
then the sum ACBand difference ABare defined by:
ACBDŒaij Cbij (add corresponding entries);
ABDŒaij bij (subtract corresponding entries):
Matrices of different sizes cannot be added or subtracted.
4. (Scalar Multiplication). If ADŒaij is a matrix and cis a scalar, then the scalar multiple cA is
defined by:
cA DŒcaij (multiply each entry by c):
5. (Matrix Multiplication). If ADŒaij is an mnmatrix and BDŒbij is a npmatrix, then
the product AB is the mpmatrix whose entries are determined as follows. To find the entry in
the row iand column jof AB:
(a) Single out row ifrom Aand column jfrom B.
(b) Multiply the corresponding entries from the row and column together and then add up the
resulting products.
Symbolically, AB DŒcij , where
cij Dai1b1j Cai2b2j CCainbnj D
n
X
kD1
aikbkj :
1
