This document defines common matrix notation and operations such as matrix addition, scalar multiplication, and matrix multiplication. It also covers properties of these operations, as well as the transpose and powers of a matrix. Key concepts include representing a matrix as a collection of columns representing vectors, defining entries and dimensions of a matrix, and rules for matrix algebra like the associative and distributive properties.

1. 2
2.1
© 2012 Pearson Education, Inc.
MATH 337
Lecture 4

2. Slide 2.1- 2© 2012 Pearson Education, Inc.
MATRIX Notation
 A is an mxn matrix
 m rows
 n columns
 scalar entry in the ith row and jth column of A is denoted
by aij and is called the (i, j)-entry of A.
 Each column of A is a list of m real numbers, which
identifies a vector in Rm
.

3. Slide 2.1- 3© 2012 Pearson Education, Inc.
MATRIX Notation
 The columns are denoted by a1, …, an, and the matrix A
is written as
A = [a1a2 … an]
 The number aij is the ith entry (from the top) of the jth
column vector aj.
 Diagonal entries in an mxn matrix are
a11, a22, a33, …, and they form the main diagonal of A.
 A diagonal matrix is a square nxn matrix whose
nondiagonal entries are zero.
 e.g., In

4. Slide 2.1- 4© 2012 Pearson Education, Inc.
Zero Matrix
 zero matrix
 All entries are zero
 written as 0
 Does not have to be square

5. Equal Matrices
 matrices are equal if
 same size (i.e., the same #of rows & columns)
 corresponding entries equal.
Slide 2.1- 5© 2012 Pearson Education, Inc.

6. Slide 2.1- 6© 2012 Pearson Education, Inc.
Adding Matrices
 If A and B are mxn matrices, then the sum A + B is
the mxn matrix whose entries are the sums of the
corresponding entries in A and B.
 The sum A + B is defined only when A and B are the
same size.
 Example 1: Let
and . Find A + B and A + C.
4 0 5 1 1 1
, ,
1 3 2 3 5 7
A B
   
= =   −   
2 3
0 1
C
− 
=  
 

7. Slide 2.1- 7© 2012 Pearson Education, Inc.
SCALAR MULTIPLES
 scalar multiple rA is the matrix whose entries are r
times the corresponding entries in A.

8. Slide 2.1- 8© 2012 Pearson Education, Inc.
Algebraic Properties of Matrices
 Theorem 1: Let A, B, and C be matrices of the
same size, and let r and s be scalars.
a. A + B = B + A
b. (A + B) + C = A + (B + C)
c. A + 0 = A
d. r(A + B) = rA + rB
e. (r + s)A = rA + sA
f. r(sA) = (rs)A

9. Matrix Multiplication - Definition
 Definition: If A is an mxn matrix, and if B is
an nxp matrix with columns b1, …, bp, then the
product AB is the mxp matrix whose columns
are Ab1, …, Abp.
 That is,
 Multiplication of matrices corresponds to
composition of linear transformations. Slide 2.1- 9© 2012 Pearson Education, Inc.
1 2 1 2
b b b b b bpp
AB A A A A   = =   L L

10. Slide 2.1- 10© 2012 Pearson Education, Inc.
MATRIX MULTIPLICATION – Row-Column Rule

11. Slide 2.1- 11© 2012 Pearson Education, Inc.
PROPERTIES OF MATRIX MULTIPLICATION
Theorem 2: Let A be an mxn matrix, and let B and C
have sizes for which the indicated sums and products
are defined.
a. (associative law of
multiplication)
b. (left distributive law)
c. (right distributive law)
d. for any scalar r
e. (identity for matrix
multiplication)
( ) ( )A BC AB C=
( )A B C AB AC+ = +
( )B C A BA CA+ = +
( ) ( ) ( )r AB rA B A rB= =
m n
I A A AI= =

12. Slide 2.1- 12© 2012 Pearson Education, Inc.
PROPERTIES OF MATRIX MULTIPLICATION
 If , we say that A and B commute with
one another.
 Warnings:
1. In general, .
2. The cancellation laws do not hold for matrix
multiplication. That is, if , then it is
not true in general that .
3. If a product AB is the zero matrix, you cannot
conclude in general that either or .
AB BA=
AB BA≠
AB AC=
B C=
0A = 0B =

13. Slide 2.1- 13© 2012 Pearson Education, Inc.
POWERS OF A MATRIX
 If A is an matrix and if k is a positive integer,
then Ak
denotes the product of k copies of A:
 If A is nonzero and if x is in Rn
then Ak
x is the result
of left-multiplying x by A repeatedly k times.
 If , then A0
x should be x itself.
 Thus A0
is interpreted as the identity matrix.
n n×
{
k
k
A A A= L
0k =

14. Slide 2.1- 14© 2012 Pearson Education, Inc.
THE TRANSPOSE OF A MATRIX
 Given an matrix A, the transpose of A is the
matrix, denoted by AT
, whose columns are
formed from the corresponding rows of A.
m n×
n m×

15. Slide 2.1- 15© 2012 Pearson Education, Inc.
Properties of Transposes
Theorem 3: Let A and B denote matrices whose sizes
are appropriate for the following sums & products.
a. (AT
)T
= A
b. (A + B)T
= AT
+ BT
c. For any scalar r, (rA)T
= rAT
d. (AB)T
= BT
AT
 The transpose of a product of matrices equals the
product of their transposes in the reverse order.

16. Chapter 1 Review – Existence Questions
 Pivot every row of A (coefficient matrix)
 For each b in Rm
, Ax=b has a solution (Thm 4)
 Each b in Rm
is a linear combination of the columns of
A (Thm 4)
 Columns of A span Rm
(Thm 4)
 T:xAx is “onto” (Thm 12)
Slide 2.1- 16© 2012 Pearson Education, Inc.

17. Chapter 1 Review – Uniqueness Questions
 Pivot every column of A (coefficient matrix)
 No free variables in corresponding system
 If solution exists, it is unique
 Only trivial solution exists to homogeneous
equation
 Columns of A are Linearly Independent
 T:xAx is “one-to-one” (Thm 12)
Slide 2.1- 17© 2012 Pearson Education, Inc.