2. Systems of Linear Equations : Recall
Chapter 3
3.1 Introduction to Linear System : p.147 to p.154
p.155 Problems: 1 to 22
3.2 Matrices and Gaussian Elimination : p.156 to p.165
p.165 Problems: 1 to 27
3.3 Reduced Row-Echelon Matrices : p.167 to p.174
p. 174 Problems: 1 to 30
3. Systems of Linear Equations : Recall
System of m equations in n unknowns (x1,…., xn)
A solution of the system must satisfy all equations at
the same time.
Two linear systems are called equivalent if they have
the same solution set.
a11x1 a12 x2 a1nxn b1
a21x1 a22 x2 a2n xn b2
am1x1 am2 x2 amn xn bm
4. Systems of Linear Equations : Recall
A system of linear equations has
1. no solution (Example 2,4 p.148,150) , or
2. exactly one solution (Examples 1,3), or
3. infinitely many solutions (Example 5, p.151).
A system of linear equations is said to be consistent
if it has either one solution or infinitely many
solutions.
A system of linear equation is said to be
inconsistent if it has no solution.
5. If , the system is called homogeneous.
A homogeneous system always has at least one solution
which is
A homogeneous system with more variables than
equations has infinitely many solutions.
b1 b2 bm 0
x1 x2 xn 0
Systems of Linear Equations : Recall
11 1 12 2 1
21 1 22 2 2
1 1 2 2
0
0
0
nn
nn
mn nm m
a x a x a x
a x a x a x
a x a x a x
6. Elementary Row Operations : Recall
The solution strategy for linear systems is to transform the
system through a series of equivalent systems until the
solution is obvious.
Elementary row operations are:
1. (Replacement) Replace one row by the sum of
itself and a multiple of another row.
2. (Interchange) Interchange two rows.
3. (Scaling) Multiply all entries in a row by a nonzero
constant.
Two matrices are called row equivalent if there is a
sequence of elementary row operations that transforms
one matrix into the other.
7. If the augmented matrices of two linear systems are
row equivalent, then the two systems have the same
solution set.
Elementary row operations guarantee that all
intermediate systems (matrices) obtained through this
process have the same set of solutions.
Any combination of elementary row operations to a
linear system yields a new linear system that is
equivalent to the first.
Elementary Row Operations : Recall
8. Matrix Notation : Recall
Augmented matrix:
An m x n matrix is a rectangular array
of numbers with m rows and n columns.
(The number of rows always comes
first.)
If m=n then the matrix is called a square
matrix.
1
2
3
3x1 matrix3x1 matrix3x3 matrix
2 1 2 10
3 1 2 1
5 4 3 4
x
x
x
2x1 x2 2x3 10
3x1 x2 2x3 1
5x1 4x2 3x3 4
2 1 2 10
3 1 2 1
5 4 3 4
9. System of m equations in n unknowns
Notation: AX = b
If b1 = b2 = … = bm = 0,
the system is homogenous
AX = 0
a11x1 a12 x2 a1nxn b1
a21x1 a22 x2 a2n xn b2
am1x1 am2 x2 amn xn bm
Matrix Notation – General Case : Recall
11 12 1 1 1
21 22 2 2 2
1 2
n
n
mn n mm m
a a a x b
a a a x b
a a a x b
1
2
11 12 1
21 22 2
1 2 m
n
n
mnm m
b
b
b
a a a
a a a
a a a
16. Basic Operations on Matrices
a) Addition
b) Scalar multiplication
c) Matrix multiplication
3.4 Matrix Operations
17. Let A be a general m x n matrix
i th row of A
j th column of A
Will sometimes write A = [ aij ]
Will sometimes write ( A )ij for aij
ai1 ai2 ain 1i m
1
2
1
j
j
mj
a
a
j n
a
3.4 Matrix Operations: General Notation
18. If m n then A is called a square matrix (same
number of equations and unknowns)
For a square matrix, the elements
a11, a22, …, ann constitute the main diagonal of A
Two matrices, A [ aij ] and B [ bij ], are equal
if they have the same dimensions and aij bij for
1 ≤ i ≤ m, 1 ≤ j ≤ n
3.4 Matrix Operations: General Notation
20. Identity matrix is a square nxn matrix In [ aij ]
where aii 1 and aij 0 for i ≠ j, i.e. the terms off
the main diagonal are all zero and the terms on the
main diagonal are all equal to 1.
4
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
I
3.4 Matrix Operations: Identity Matrix
3
1 0 0
0 1 0
0 0 1
I 2
1 0
0 1
I
1
1I
22. Adding matrices means adding corresponding
elements, e.g.
Let A = [ aij ] and B = [ bij ] be two m x n matrices,
then C = A + B is a m x n matrix C = [ cij ] such that
cij = aij + bij for all i, j
Note: sizes of matrices must be the same
1 2 3 3 2 1 4 4 4
4 5 6 6 5 4 10 10 10
1 2 1 2 3
3 4 4 5 6
undefined
3.4 Matrix Operations: Matrix Addition
23. Scalar multiplication means multiplying each element
of a matrix by the same scalar, e.g.
Let A = [ aij ] and r R
Then C = r A, where
C = [ cij ] is defined as cij = r aij i, j
1 2 2 4
2
3 4 6 8
3.4 Matrix Operations: Scalar Multiplication
4 1 3 8 2 6
2
2 5 0 4 10 0
25. See p. 181 Examples 5 & 6
3.4 Matrix Operations: Matrix Multiplication
26. We want to express
So, need to have matrix multiplication to work like
e.g.
11 12 111 1 12 2 1 1
21 22 221 1 22 2 2 2
as
a a xa x a x b b
a a xa x a x b b
11 12 1 11 1 12 2
21 22 2 21 1 22 2
a a x a x a x
a a x a x a x
1 2 5 6 1 5 2 7 1 6 2 8 19 22
3 4 7 8 3 5 4 7 3 6 4 8 43 50
3.4 Matrix Operations: Matrix Multiplication
27. If A is m x n and X is n x 1, can also express AX as
We call this a linear combination of the columns of A.
The coefficients are the elements of X.
11 12 1
21 22 2
1 2
1 2
n
n
n
mnm m
a a a
a a a
x x x
a a a
AX
3.4 Matrix Operations: Matrix Multiplication
28. What about
The basic idea is to multiply each column of B by A
11 12 111 12 1
21 22 2 21 22 2
1 2 1 2
pn
n p
mnm m npn n
b b ba a a
a a a b b b
a a a b b b
A B
111 12
221 22
1 2
p
p
n n np
bb b
bb b
b b b
AB A A A
3.4 Matrix Operations: Matrix Multiplication
29. Examine the first column of product
1 1
1
11
2 121
1
1
1
1
n
k k
k
n
k k
k
n n
mk k
k
a b
b
a bb
b
a b
A
1st row A x 1st column B
2nd row A x 1st column B
mth row A x 1st column B
3.4 Matrix Operations: Matrix Multiplication
30. Defn. Let A = [ aij ] be an m x n matrix and let
B = [ bij ] be an q x p matrix.
The product of A and B is defined if and only if n=q
Then, AB = C = [ cij ], is the m x p matrix defined by
The product of B and A is defined if and only if m=p
Then, BA = D = [ dij ], is the q x n matrix defined by
1 1 2 2
1
for 1,2, , 1,2, ,
n
ij in nji j i jik kj
k
c a b a b a b a b i m j p
3.4 Matrix Operations: Matrix Multiplication
1 1 2 2
1
for 1,2, , 1,2, ,
m
ij in nji j i jik kj
k
d b a b a b a b a i q j n
31. 3.4 Matrix Addition: Properties
Let A, B and C be m x n matrices
1) A + B is an m x n matrix
2) A + (B + C) = (A + B) + C
3) There is a unique m x n matrix m0n such that
A + m0n = m0n + A = A for every matrix A
4) For every m x n matrix A, there is a unique m x n
matrix D such that A + D = D + A = m0n
5) A + B = B + A
32. 3.4 Matrix Multiplication: Properties
If A, B and C are matrices of the appropriate sizes, then
A(BC) (AB)C
If A, B and C are matrices of the appropriate sizes, then
(A + B)C AC + BC
If A, B and C are matrices of the appropriate sizes, then
C(A + B) CA + CB
Let A be any mxn matrix,
then A In A and Im A A
Let A be any mxn matrix,
then A 0nxp 0mxp and 0qxm A 0qxn
33. Comments on Matrix Multiplication
AB need not equal BA
Let
So, we can have AB 0, but A ≠ 0 and B ≠ 0
BA = ?
1 2 4 6 0 0
2 4 2 3 0 0
A B AB
1 2 2 1 2 7
Let , ,
2 4 3 2 5 1
8 5
then but , i.e. can't cancel
16 10
A B C
AB AC B C
3.4 Matrix Multiplication: Properties
34. Example 8 (p.185) 51 3 4
14 1 2 7 3 2 1
43 1 1 5 2 2 3
32 1 3
25 10
and
18 5
A B C
AB AC B C
3.4 Matrix Multiplication: Properties
2 1
4 1 2 7 1 2 0 0
3 1 1 5 0 1 0 0
1 0
and , are non zero matrices
A D AD
A D
36. If r and s are real numbers and A and B are matrices,
then
r ( sA ) ( rs ) A
( r + s ) A rA + sA
r ( A + B ) rA + rB
A ( rB ) r ( AB ) ( rA ) B
3.4 Scalar Multiplication: Properties
37. Let A [ aij ] be an m x n matrix.
The transpose of A, AT [ aij
T ], is the n x m matrix
defined by aij
T aji
T1 2 1 3
3 4 2 4
A A
3.4 Matrix Operations: Matrix Transpose
T 3
3 5
5
B B
T
1 2
1 3 4
3 5
2 5 6
4 6
C C
38. If r is a scalar and A and B are matrices, then
( AT )T A
( A + B )T AT + BT when AT A
( AB )T BTAT A is called symmetric
( rA )T rAT
Find transpose matrix for:
3.4 Matrix Transpose: Properties
2 3
1 4
5 6
3 5 2
5 4 1
2 1 7
1 3 4
39. The following laws of exponents hold for nonnegative
integers p and q and any n x n matrix A
1 ) Ap Aq Ap + q
2 ) ( Ap ) q Apq
Caution. Without additional assumptions on A and B,
cannot do the following
1 ) define Ap for negative integers p
2 ) assert that ( AB ) p Ap B p
3.4 Matrix Powers: Properties
40. An n x n matrix A [ aij ] is called upper triangular
if and only if aij 0 for i > j
An n x n matrix A [ aij ] is called lower triangular
if and only if aij 0 for i < j
Note:
A diagonal matrix is both upper and lower triangular
The n x n zero matrix is both upper and lower triangular
3.4 Special Matrices: Triangular Matrices
41. Defn - A matrix A is called symmetric if AT A
Defn - A matrix A is called skew-symmetric if AT A
Comment - If A is skew-symmetric, then the diagonal
elements of A are zero
Comment - Any square matrix A can be written as the
sum of a symmetric matrix and a skew-symmetric matrix
T T
symmetric skew-symmetric
1 1
2 2
A A A A A
3.4 Special Matrices: Symmetric Matrices
42. Symmetry
For any matrix A, matrices AAT and AT A
are symmetric.
If matrix A is square, then A+ AT is symmetric.
3.4 Special Matrices: Symmetric Matrices
45. Nonsingular Matrices
Defn - An n x n matrix A is called nonsingular or
invertible if there exists an n x n matrix B such that
AB BA In
Comments
If B exists, then B is called the inverse of A.
Then, A is also the inverse of B.
If B does not exist, then A is called singular or
noninvertible.
At this point, the only available tool for showing that
A is nonsingular is to show that B exists
3.5 Inverse of Matrices : Definition
3 5 2 5
1 2 1 3
A B
46. Nonsingular Matrices
Theorem - If the inverse of a matrix exists, then that
inverse is unique.
Proof - Let A be a nonsingular n x n matrix and let B and
C be inverses of A. Then AB BA In and
AC CA In B B In B( AC ) ( BA )C In C C
so the inverse is unique.
Notation - If A is a nonsingular matrix, the inverse of A is
denoted by A1
Comment - For nonsingular matrices, A, can define A
raised to a negative power as Ak ( A1 ) k k > 0
3.5 Inverse of Matrices : Properties
47. Nonsingular Matrices
Theorem - If A and B are both nonsingular matrices, then
the product AB is nonsingular and ( AB ) 1 B1A1
Proof - Consider the following products
AB ( B1A1 ) AB B1A1 A In A1 AA1 In
and
( B1A1 ) AB B1A1AB B1In B B1B In
Since we have found a matrix C such that C ( AB ) ( AB )
C In , AB is nonsingular and its inverse is C B1A1
3.5 Inverse of Matrices : Properties
48. Nonsingular Matrices
Theorem - If A1, A2, …, Ar are nonsingular matrices,
then A1 A2 • • • Ar is nonsingular and
Theorem - If A is a nonsingular matrix, then A1 is
nonsingular and (A1)1 A
Proof: Since A1A AA1 In , then A1 is nonsingular
and its inverse is A. So (A1)1 A.
A1A2 Ar
1
Ar
1
Ar1
1
A2
1
A1
1
3.5 Inverse of Matrices : Properties
49. Nonsingular Matrices
Comment: We have observed earlier that AB AC does
not necessarily imply that B C. However, if A is an n x n
nonsingular matrix and AB AC, then B C.
A1 ( AB ) A1 ( AC ) ( A1A ) B ( A1A ) C B C
Comment: We have observed earlier that AB n0n does not
imply that A n0n or B n0n. However, if A is an n x n
nonsingular matrix and AB n0n , then B n0n .
A1 ( AB ) A1
n0n ( A1A ) B n0n B n0n
3.5 Inverse of Matrices : Properties
50. Nonsingular Matrices
Theorem - If A is a nonsingular matrix, then AT is
nonsingular and ( AT ) 1 ( A1 ) T
Proof - By an earlier theorem, ( AB )T BTAT for
any two matrices A and B. Since A is nonsingular,
A1 A A A1 In . Applying the relationship on
transposes gives
In In
T ( A1 A )T AT ( A1 )T
In In
T ( AA1 )T ( A1 )T AT
Since AT ( A1 )T In and ( A1 )T AT In , AT is
nonsingular and its inverse is ( A1 )T , i.e.
( AT ) 1 ( A1 ) T
3.5 Inverse of Matrices : Properties
51. Linear Systems and Inverses
A system of n linear equations in n unknowns may be
written as AX = b, where A is n x n matrix. If A is
nonsingular, then A1 exists and the system may be
solved by multiplying both sides by A1
A1( AX ) = A1b ( A1A )X = A1b X = A1b
If A is an invertible matrix, then AX = b has a unique
solution for any b.
3.5 Inverse of Matrices : Properties
52. Linear Systems and Inverses
Comment - Although X = A1B gives a simple
expression for the solution, its primary usage is for
proofs and derivations
At this point we have no practical tool for computing
A1
Even with a tool for computing A1, this method of
solution is usually numerically inefficient. The only
exception is if A has a special structure that lets A1
have a simple relationship to A
3.5 Inverse of Matrices : Properties
53. Linear Systems and Inverses
AX = b
If then
For k non zero scalar and A an invertible matrix
3.5 Inverse of Matrices : Properties
a b
c d
A
0ad bc
1 1 d b
ad bc c a
A
1 11
k
k
A A
54. 3.5 Inverse of Matrices : Simple Cases
1 1
n n n
A I A I I
1
1
0 0 0 0
10 0 0 0 0 0 0 0
0 0 0 0
1
, , , , , 00 0 0 0 0 0 0 0
0 0 0 0
1
0 0 0 00 0 0 0
1
0 0 0 0
a
a
bb
a b c d ec
c
d
e d
e
D D