matrices and algbra

Systems of Linear
Equation and Matrices
CHAPTER 1
FASILKOM UI 05

Linear Algebra - Chapter 1 [YR2005] 3
Introduction ~ Matrices
 Information in science and mathematics is
often organized into rows and columns to
form rectangular arrays.
 Tables of numerical data that arise from
physical observations
 Example: (to solve linear equations)
 Solution is obtained by performing appropriate
operations on this matrix






− 412
315
42
35
=−
=+
yx
yx

1.1 Introduction to
Systems of Linear Equations

Linear Equations
 In x y variables (straight line in the xy-plane)
where a1, a2, & b are real constants,
 In n variables
where a1, …, an & b are real constants
x1, …, xn = unknowns.
 Example 1 Linear Equations
 The equations are linear (does not involve any
products or roots of variables).
byaxa =+ 21
bxaxaxa nn =+++ ...2211
73 =+ yx 132
1
++= zxy 732 4321 =+−− xxxx

Linear Equations
 The equations are not linear.
 A solution of is a sequence
of n numbers s1, s2, ..., sn Э they satisfy the equation
when x1=s1, x2=s2, ..., xn=sn (solution set).
 Example 2 Finding a Solution Set
 1 equation and 2 unknown, set one var as the
parameter (assign any value)
 or
 1 equation and 3 unknown, set 2 vars as parameter
53 =+ yx 423 =+−+ xzzyx xy sin=
bxaxaxa nn =+++ ...2211
124 =− yx
2
1
2, −== tytx tytx =+= ,4
1
2
1
574 321 =+− xxx
txsxtsx ==−+= 321 ,,745

Linear Systems / System of
Linear Equations
 Is A finite set of linear equations in the vars x1, ..., xn
 s1, ..., sn is called a solution if x1=s1, ..., xn=sn is a solution
of every equation in the system.
 Ex.
 x1=1, x2=2, x3=-1 the solution
 x1=1, x2=8, x3=1 is not, satisfy only the first eq.
 System that has no solution : inconsistent
 System that has at least one solution: consistent
 Consider:
493
134
321
321
−=++
−=+−
xxx
xxx
00,:
00,:
112222
111111
≠∧≠=+
≠∧≠=+
bacybxal
bacybxal

 (x,y) lies on a line if and only if the numbers x and y satisfy
the equation of the line. Solution: points of intersection l1 & l2
 l1 and l2 may be parallel:
no intersection, no solution
 l1 and l2 may intersect
at only one point: one solution
 l1 and l2 may coincide:
infinite many points of intersection,
infinitely many solutions
Linear Systems
x
x
y1l 2l
x
y
x
y
1l 2l
21 & ll

Linear Systems
 In general: Every system of linear equations has
either no solutions, exactly one solution, or infinitely
many solutions.
 An arbitrary system of m linear equations in n
unknowns:
a11x1 + a12x2 + ... + a1nxn = b1
a21x1 + a22x2 + ... + a2nxn = b2
am1x1 + am2x2 + ... + amnxn = bm
 x1, ..., xn = unknowns, a’s and b’s are constants
 aij, i indicates the equation in which the coefficient
occurs and j indicates which unknown it multiplies

  

Augmented Matrices
 Example:
 Remark: when constructing, the unknowns must be
written in the same order in each equation and the
constants must be on the right.












mmnmm
n
n
baaa
baaa
baaa




21
222221
111211
0563
1342
92
321
321
321
=−+
=−+
=++
xxx
xxx
xxx










−
−
0563
1342
9211

Augmented Matrices
 Basic method of solving system linear equations
 Step 1: multiply an equation through by a nonzero
constant.
 Step 2: interchange two equations.
 Step 3: add a multiple of one equation to another.
 On the augmented matrix (elementary row
operations):
 Step 1: multiply a row through by a nonzero
constant.
 Step 2: interchange two rows.
 Step 3: add a multiple of one equation to another.

Elementary Row Operations
(Example)
 r2= -2r1 + r2
 r3 = -3r1 + r3
0563:
1772:
92:
3
2
1
=−+
−=−
=++
zyxr
zyr
zyxr
0563:
1342:
92:
3
2
1
=−+
=−+
=++
zyxr
zyxr
zyxr










−
−
0563
1342
9211










−
−−
0563
17720
9211
27113:
1772:
92:
3
2
1
−=−
−=−
=++
zyr
zyr
zyxr










−−
−−
271130
17720
9211

(Example)
 r2 = ½ r2
 r3 = -3r2 + r3

27113:
:
92:
3
2
17
2
7
2
1
−=−
−=−
=++
zyr
zyr
zyxr










−−
−−
271130
10
9211
2
17
2
7
2
3
2
1
3
2
17
2
7
2
1
:
:
92:
−=−
−=−
=++
zr
zyr
zyxr










−−
−−
2
3
2
1
2
17
2
7
00
10
9211
3:
:
92:
3
2
17
2
7
2
1
=
−=−
=++
zr
zyr
zyxr
7 17
2 2
1 1 2 9
0 1
0 0 1 3
 
 − − 
  

(Example)
 r1 = r1 – r2
 r1 = -11/2 r3 + r1
 r2 = 7/2 r3 + r2
 Solution:
3:
:
:
3
2
17
2
7
2
2
35
2
11
1
=
−=−
=+
zr
zyr
zxr










−−
3100
10
01
2
17
2
7
2
35
2
11










3100
2010
1001
3:
2:
1:
3
2
1
=
=
=
zr
yr
xr
3:
:
1:
3
2
17
2
7
2
1
=
−=−
=
zr
zyr
xr










−−
3100
10
1001
2
17
2
7

Echelon Forms
 Reduced row-echelon form, a matrix must have
the following properties:
 If a row does not consist entirely of zeros the the
first nonzero number in the row is a 1 = leading 1
 If there are any rows that consist entirely of zeros,
then they are grouped together at the bottom of
the matrix.
 In any two successive rows that do not consist
entirely of zeros, the leading 1 in the lower row
occurs farther to the right than the leading 1 in the
higher row.
 Each column that contains a leading 1 has zeros
everywhere else.

Echelon Forms
 A matrix that has the first three properties is
said to be in row-echelon form.
 Example:
 Reduced row-echelon form:
 Row-echelon form:

















 −




















−
00
00
,
00000
00000
31000
10210
,
100
010
001
,
1100
7010
4001
1 4 3 7 1 1 0 0 1 2 6 0
0 1 6 2 , 0 1 0 , 0 0 1 1 0
0 0 1 5 0 0 0 0 0 0 0 1
−     
     −     
          

Elimination Methods
 Step 1: Locate the
leftmost non zero
column
 Step 2: Interchange
r2 ↔ r1.
 Step 3: r1 = ½ r1.
 Step 4: r3 = r3 – 2r1.










−−−
−
−
156542
281261042
1270200










−−−
−
−
156542
1270200
281261042










−−−
−
−
156542
1270200
1463521










−−
−
−
29170500
1270200
1463521

Elimination Methods
 Step 5 : continue do all steps above until the entire
matrix is in row-echelon form.
 r2 = -½ r2
 r3 = r3 – 5r2
 r3 = 2r3










−−
−−
−
29170500
60100
1463521
2
7










−−
−
10000
60100
1463521
2
1
2
7










−−
−
210000
60100
1463521
2
7

Elimination Methods
 Step 6 : add suitable multiplies of each row to the
rows above to introduce zeros above the leading 1’s.
 r2 = 7/2 r3 + r2
 r1 = -6r3 + r1
 r1 = 5r2 + r1









 −
210000
100100
1463521









 −
210000
100100
203521










210000
100100
703021

Elimination Methods
 1-5 steps produce a row-echelon form
(Gaussian Elimination). Step 6 is producing
a reduced row-echelon (Gauss-Jordan
Elimination).
 Remark: Every matrix has a unique reduced
row-echelon form, no matter how the row
operations are varied. Row-echelon form of
matrix is not unique: different sequences of
row operations can produce different row-
echelon forms.

Back-substitution
 Bring the augmented matrix into row-echelon form
only and then solve the corresponding system of
equations by back-substitution.
 Example: [Solved by back substitution]











 −
0000000
100000
1302100
0020231
3
1
3
1
6
643
5321
132
0223
=
=++
=+−+
x
xxx
xxxx
3
1
6
643
5321
321
2231.Step
=
−−=
−+−=
x
xxx
xxxx

Back-Substitution
3
1
6
43
5421
43
3
1
6
43
5321
3
1
6
2
243
2ngSubstituti
2
223
ngSubstituti
2.Step
=
−=
−−−=
−=
=
−=
−+−=
=
x
xx
xxxx
xx
x
xx
xxxx
x
Step 3. Assign arbitrary values
to the free variables
[parameters], if any
3
1
6
5
4
3
2
1
2
243
=
=
=
−=
=
−−−=
x
tx
sx
sx
rx
tsrx

Homogeneous Linear Systems
 A system of linear equations is said to be
homogeneous if the constant terms are all zero.
 Every homogeneous sytem of linear equations is
consistent, since all such systems have
x1=0,x2=0,...,xn=0 as a solution [trivial solution]. Other
solutions are called nontrivial solutions.
01
0
0
221
2222121
1212111
=+++
=+++
=+++
nmnmm
nn
nn
xaxaxa
xaxaxa
xaxaxa





 Example: [Gauss-Jordan Elimination]
0
02
032
022
543
5321
54321
5321
=++
=−−+
=+−+−−
=+−+
xxx
xxxx
xxxxx
xxxx












⇒












−−
−−−
−
000000
001000
010100
010011
011100
010211
013211
010122

The corresponding system of equations is
Solving for the leading variables yields
The general solution is
The trivial solution is obtained when s=t=0
04
53
521
=
−=
−−=
x
xx
xxx
0
0
0
4
53
521
=
=+
=++
x
xx
xxx
txxtxsxtsx ==−==−−= 54321 ,0,,,

 Theorem:
A homogeneous system of linear equations
with more unknowns than equations has
infinitely many solutions.

1.3 Matrices and Matrix Operations

Matrix Notation and Terminology
 A matrix is a rectangular array of numbers with rows
and columns.
 The numbers in the array are called the entries in the
matrix.
 Examples:
 The size of a matrix is described in terms of the
number of rows and columns its contains.
 A matrix with only one column is called a column
matrix or a column vector.
 A matrix with only one row is called a row matrix
or a row vector.
[ ] [ ]4,
3
1
,
000
10
2
,3012,
41
03
21
2
1















 −
−










−
πe

Matrix Notation and Terminology
 aij = (A)ij = the entry in row i and column j of a
matrix A.
 1 x n row matrix a = [a1 a2 ... an]
 m x 1 column matrix
 A matrix A with n rows and n columns is called
a square matrix of order n. Main diagonal of A
= {a11, a22, ..., ann}












=
nb
b
b
b

2
1












nnnn
n
n
aaa
aaa
aaa




21
22221
11211

Operations on Matrices
 Definition:
 Two matrices are defined to be equal if they have the
same size and their corresponding entries are equal.
 If A = [aij] and B = [bij] have the same size, then A=B if
and only if (A)ij=(B)ij, or equivalently aij=bij for all i and j.
 Definition:
 If A and B are matrices of the same size, then the sum
A+B is the matrix obtained by adding the entries of B to
the corresponding entries of A, and the difference A–B
is the matrix obtained by subtracting the entries of B
from the corresponding entries of A. Matrices of
different sizes cannot be added or subtracted.

 If A = [aij] and B = [bij] have the same size, then
(A+B)ij = (A)ij + (B)ij = aij + bij and
(A-B)ij = (A)ij – (B)ij = aij - bij
 Definition:
 If A is any matrix and c is any scalar, then the product
cA is the matrix obtained by multiplying each entry of
the matrix A by c. The matrix cA is said to be a scalar
multiple of A.
 If A = [aij], then (cA)ij = c(A)ij = caij.

 Definition
If A is an mxr matrix and B is an rxn matrix,
then the product AB is the mxn matrix whose
entries are determined as follows. To find the
entry in row i and column j of AB, single out
row i from the matrix A and column j from the
matrix B. Multiply the corresponding entries
from the row and column together and then
add up the resulting products.

Partitioned Matrices

Matrix Multiplication
by Columns and by Rows

Matrix Products as Linear Combinations

Matrix Form of a Linear System

Transpose of a Matrix

1.4 Inverses; Rules of Matrix Arithmetic

Properties of Matrix Operations
 ab = ba for real numbers a & b, but AB ≠ BA
even if both AB & BA are defined and have
the same size.
 Example:






−
=




 −−
=






=




−
=
03
63
,
411
21
03
21
,
32
01
BAAB
BA

 Theorem: Properties of
a) A+B = B+A
b) A+(B+C) = (A+B)+C
c) A(BC) = (AB)C
d) A(B+C) = AB+AC
e) (B+C)A = BA+CA
f) A(B-C) = AB-AC
g) (B-C)A = BA-CA
h) a(B+C) = aB+aC
i) a(B-C) = aB-aC
Math Arithmetic
(Commutative law for addition)
(Associative law for addition)
(Associative for multiplication)
(Left distributive law)
(Right distributive law)
j) (a+b)C = aC+bC
k) (a-b)C = aC-bC
l) a(bC) = (ab)C
m) a(BC) = (aB)C

 Proof (d):
 Proof for both have the same size:
 Let size A be r x m matrix, B & C be m x n (same
size).
 This makes A(B+C) an r x n matrix, follows that
AB+AC is also an r x n matrix.
 Proof that corresponding entries are equal:
 Let A=[aij], B=[bij], C=[cij]
 Need to show that [A(B+C)]ij = [AB+AC]ij for all values
of i and j.
 Use the definitions of matrix addition and matrix
multiplication.

 Remark: In general, given any sum or any product of
matrices, pairs of parentheses can be inserted or
deleted anywhere within the expression without
affecting the end result.
ij
ijij
mjimjijimjimjiji
mjmjimjjijjiij
ACAB
acAB
cacacabababa
cbacbacbaCBA
][
][][
)()(
)()()()]([
22112211
222111
+=
+=
+++++++=
++++++=+



Zero Matrices
 A matrix, all of whose entries are zero, such as
 A zero matrix will be denoted by 0 or 0mxn for the mxn zero
matrix. 0 for zero matrix with one column.
 Properties of zero matrices:
 A + 0 = 0 + A = A
 A – A = 0
 0 – A = -A
 A0 = 0; 0A = 0
[ ]0,
0
0
0
0
,
0000
0000
,
000
000
000
,
00
00



































Identity Matrices
 Square matrices with 1’s on the main diagonal and
0’s off the main diagonal, such as
 Notation: In = n x n identity matrix.
 If A = m x n matrix, then:
 AIn = A and InA = A




























1000
0100
0010
0001
,
100
010
001
,
10
01

Identity Matrices
 Example:
 Theorem: If R is the reduced row-echelon form of an
n x n matrix A, then either R has a row of zeros or R
is the identity matrix In.






=
232221
131211
aaa
aaa
A
A
aaa
aaa
aaa
aaa
AI =





=











=
232221
131211
232221
131211
2
10
01
A
aaa
aaa
aaa
aaa
AI =





=
















=
232221
131211
232221
131211
3
100
010
001

Identity Matrices
 Definition: If A & B is a square matrix and same size
Э AB = BA = I, then A is said to be invertible and B is
called an inverse of A. If no such matrix B can be
found, then A is said to be singular.
 Example:






−
−
=





=
31
52
,
21
53
AB
IBA
IAB
=





=





−
−






=
=





=











−
−
=
10
01
31
52
21
53
10
01
21
53
31
52

Properties of Inverses
 Theorem:
 If B and C are both inverses of the matrix A, then B = C.
 If A is invertible, then its inverse will be denoted by the
symbol A-1
.
 The matrix
is invertible if ad-bc ≠ 0, in which case the inverse is
given by the formula






=
dc
ba
A










−−
−
−
−
−=





−
−
−
=−
bcad
a
bcad
c
bcad
b
bcad
d
ac
bd
bcad
A
11

Properties of Inverses
 Theorem: If A and B are invertible matrices of the
same size, then AB is invertible and (AB)-1
= B-1
A-1
.
 A product of any number of invertible matrices is
invertible, and the inverse of the product is the
product of the inverses in the reverse order.
 Example:






=





=





=
89
67
,
22
23
,
31
21
ABBA








−
−
=





−
−
=





−
−
= −−−
2
7
2
9
34
)(,
1
11
,
11
23 1
2
3
11
ABBA








−
−
=





−
−






−
−
=−−
2
7
2
9
34
11
23
1
11
2
3
11
AB

Powers of a Matrix
 If A is a square matrix, then we define the nonnegative integer
powers of A to be
A0=I An
= AA...A (n>0)
n factors
 Moreover, if A is invertible, then we define the negative integer
prowers to be A-n
= (A-1
)n
= A-1
A-1
...A-1
n factors
 Theorem: Laws of Exponents
 If A is a square matrix, and r and s are integers, then Ar
As
=
Ar+s
= Ars
 If A is an invertible matrix, then
 A-1
is invertible and (A-1
)-1
= A
 An
is invertible and (An
)-1
= (A-1
)n
for n = 0, 1, 2, ...
 For any nonzero scalar k, the matrix kA is invertible and
(kA)-1
= 1/k A-1
.

Powers of a Matrix
 Example:






=
31
21
A 





−
−
=−
11
231
A






=

















=
4115
3011
31
21
31
21
31
213
A






−
−
=





−
−






−
−






−
−
== −−
1115
3041
11
23
11
23
11
23
)( 313
AA

Polynomial Expressions Involving
Matrices
 If A is a square matrix, m x m, and if
is any polynomial, then we define
 Example:
n
n xaxaaxp +++= 10)(
n
n AaAaIaAp +++= 10)(
432)( 2
+−= xxxp






=





+




−
−





=






+




−
−




−
=+−=
130
29
40
04
90
63
180
82
10
01
4
30
21
3
30
21
2432)(
2
2
IAAAp

Properties of the Transpose
 Theorem: If the sizes of the matrices are
such that the stated operations can be
performed, then
a) ((A)T
)T
= A
b) (A+B)T
= AT
+ BT
and (A-B)T
= AT
– BT
c) (kA)T
= kAT
, where k is any scalar
d) (AB)T
= BT
AT
 The transpose of a product of any number of
matrices is equal to the product of their
transpose in the reverse order.

Invertibility of a Transpose
 Theorem: If A is an
invertible matrix, then AT
is also invertible and
(AT
)-1
= (A-1
)T
 Example:






−
−
=





−
−
=





−−
=






−
−
=




 −−
=
−−−
53
21
)(,
53
21
)(,
52
31
13
25
,
12
35
111 TT
T
AAA
AA

Exercise
 Show that if a square matrix A satisfies A2
-3A+I=0,
then A-1
=3I-A
 Let A be the matrix
Determine whether A is invertible, and if so, find its
inverse. [Hint. Solve AX = I by equating
corresponding entries on the two sides.]










110
011
101

1.5 Elementary Matrices and
a Method for Finding A-1

Elementary Matrices
 Definition:
 An n x n matrix is called an elementary matrix if
it can be obtained from the n x n identity matrix In
by performing a single elementary row
operation.
 Example:
1. Multiply the second row of I2 by -3.
2. Interchange the second and fourth rows of I4.
3. Add 3 times the third row of I3 to the first row.




























−
100
010
301
:3,
0010
0100
1000
0001
:2,
30
01
:1

Elementary Matrices
 Theorem: (Row Operations by Matrix Multiplication)
 If the elementary matrix E results from performing a certain
row operation on Im and if A is an m x n matrix, then the
product of EA is the matrix that results when this same row
operation is performed on A.
 Example:
 EA is precisely the same matrix that results when we add 3
times the first row of A to the third row.










−=










=










−=
01044
6312
3201
103
010
001
,
0441
6312
3201
EA
EA

Elementary Matrices
 If an elementary row operation is applied to an
identity matrix I to produce an elementary matrix E,
then there is a second row operation that, when
applied to E, produces I back again.
 Inverse operation
Row operation on I that
produces E
Row operation on E that
reproduces I
Multiply row i by c ≠ 0 Multiply row i by 1/c
Interchange rows i and j Interchange rows i and j
Add c times row i to row j Add –c times row i to row j

Elementary Matrices
 Theorem: Every elementary matrix is invertible,
and the inverse is also an elementary matrix.
 Theorem: (Equivalent Statements)
 If A is an n x n matrix, then the following
statements are equivalent, that is, all true or all
false.
a) A is invertible
b) Ax = 0 has only the trivial solution.
c) The reduced row-echelon form of A is In.
d) A is expressible as a product of elementary
matrices.

Elementary Matrices
 Proof:
Assume A is invertible and let x0 be any solution of Ax=0.
Let Ax=0 be the matrix form of the system
)()()()()( adcba ⇒⇒⇒⇒
)()( ba ⇒
0,0,0)(,0)( 000
11
0
1
==== −−−
xIxxAAAAxA
)()( cb ⇒
0
0
11
1111
=++
=++
nnnn
nn
xaxa
xaxa



















⇒












010000
00100
00010
00001
0
0
0
21
22221
11211








nnnn
n
n
aaa
aaa
aaa

Elementary Matrices
Assumed that the reduced row-echelon form of A is In by a finite
sequence of elementary row operations, such that:
By theorem, E1,…,En are invertible. Multiplying both sides of equation
on the left we obtain:
This equation expresses A as a product of elementary matrices.
If A is a product of elementary matrices, then the matrix A is a product
of invertible matrices, and hence is invertible.
 Matrices that can be obtained from one another by a finite
sequence of elementary row operations are said to be row
equivalent.
 An n x n matrix A is invertible if and only if it is row equivalent to
the n x n identity matrix.
)()( dc ⇒
nk IAEEE =12
11
2
1
1
11
2
1
1
−−−−−−
== knk EEEIEEEA 
)()( ad ⇒

A Method for Inverting Matrices
 To find the inverse of an invertible matrix, we must
find a sequence of elementary row operations that
reduces A to the identity and then perform this same
sequence of operations on In to obtain A-1
.
 Example:
 Adjoin the identity matrix to the right side of A,
thereby producing a matrix of the form [A|I]
 Apply row operations to this matrix until the left side
is reduced to I, so the final matrix will have the form
[I|A-1].










=
801
352
321
A

Added –2 times the first row to the
second and –1 times the first row to the
third.
Added 2 times the second row to the
third.
Multiplied the third row by –1.
Added 3 times the third row to the
second and –3 times the third row to the
first.
We added –2 times the second row to
the first.





−−
−−
−










−−
−−
−










−−
−





−





−
−





−
−





−
−





−
−










125
3513
91640
100
010
001
125
3513
3614
100
010
021
125
012
001
100
310
321
125
012
001
100
310
321
101
012
001
520
310
321
100
010
001
801
352
321

 Often it will not be known in advance whether a given matrix is
invertible.
 If elementary row operations are attempted on a matrix that is
not invertible, then at some point in the computations a row of
zeros will occur on the left side.
 Example:
Added -2 times the first row to the second and
added the first row to the third.
Added the second row to the third.










−−
=
521
142
461
A










−
−−−










−−−










−−
111
012
001
000
980
461
101
012
001
980
980
461
100
010
001
521
142
461

Exercises
 Consider the matrices
Find elementary matrices, E1, E2, E3, and E4, such
that
a. E1A=B
b. E2B=A
c. E3A=C
d. E4C=A










−
−−=










−−=










−−=
372
172
143
,
143
172
518
,
518
172
143
CBA

Exercises
 Express the matrix:
in the form A = E F G R, where E, F, G are
elementary matrices, and R is in row-echelon form.










−−− 8152
8331
8710

1.6 Further Results on
Systems of Equations and Invertibility

Linear Systems
 Theorem:
Solving Linear Systems by Matrix Inversion:
If A is an invertible n x n matrix, then for each n x 1
matrix b, the system of equations Ax = b has exactly
one solution, namely, x = A-1
b.
 Linear systems with a common coefficient matrix.
Ax=b1, Ax=b2, Ax=b3, ..., Ax=bk
If A is invertible, then the solutions
x1=A-1
b1, x2=A-1
b2, x3=A-1
b3, ..., xk=A-1
bk
This can be efficiently done using Gauss-Jordan
Elimination on [A|b1|b2|...|bk]

Linear Systems
 Example: (a) (b)
The solution:
(a) x1=1, x2=0, x3=1
(b) x1=2, x2=1, x3=-1
98
5352
432
31
321
321
=+
=++
=++
xx
xxx
xxx
68
6352
132
31
321
321
−=+
=++
=++
xx
xxx
xxx










−










−
1
1
2
1
0
1
100
010
001
6
6
1
9
5
4
801
352
321

Properties of Invertible Matrices
 Theorem: Let A be a square matrix.
a) If B is a square matrix satisfying BA=I, then B=A-1
.
b) If B is a square matrix satisfying AB=I, then B=A-1
.
 Theorem: Equivalent Statements
a) A is invertible
b) Ax=0 has only the trivial solutions
c) The reduced row-echelon form of A is In
d) A is expresssible as a product of elementary matrices
e) Ax=b is consistent for every n x 1 matrix b
f) Ax=b has exactly one solution for every n x 1 matrix b

Properties of Invertible Matrices
 Theorem: Let A and B be square matrices of
the same size. If AB is invertible, then A and
B must also be invertible.
 A fundamental problem.
Let A be a fixed m x n matrix. Find all m x 1
matrices b such that the system of equations
Ax=b is consistent.

Exercises
 Solve the system by inverting the coefficient matrix.
 Find condition that b’s must satisfy for the system to
be consistent.
6642
4973
744
032
=−−−−
=+++
=+++
=−−−
zyxw
zyxw
zyxw
zyx
221
121
23
46
bxx
bxx
=−
=−

1.7 Diagonal, Triangular,
and Symmetric Matrices

Diagonal Matrices
 A square matrix in which all the entries off the main
diagonal are zero. Example:
 A diagonal matrix is invertible if and only if all of its
diagonal entries are nonzero.












−
















−
8000
0000
0040
0006
,
100
010
001
,
50
02












=
nd
d
d
D




00
00
00
2
1
















=−
nd
d
d
D
100
010
001
2
1
1


















=
k
n
k
k
k
d
d
d
D




00
00
00
2
1

Diagonal Matrices
 Example:










−=
200
030
001
A
















−=−
2
1
00
0
3
1
0
001
1
A










−=
200
030
001
5
A










−=−
32
1
243
15
00
00
001
A










=






























=




















333322311
233222211
133122111
3
2
1
333231
232221
131211
343333323313
242232222212
141131121111
34333231
24232221
14131211
3
2
1
00
00
00
00
00
00
adadad
adadad
adadad
d
d
d
aaa
aaa
aaa
adadadad
adadadad
adadadad
aaaa
aaaa
aaaa
d
d
d

Triangular Matrices
 Lower triangular = a square matrix in which all the
entries above the main diagonal are zero.
 Upper triangular = a square matrix in which all the
entries under the main diagonal are zero.
 Triangular = a matrix that is either upper triangular or
lower triangular.












44
3433
242322
14131211
000
00
0
a
aa
aaa
aaaa












44434241
333231
2221
11
0
00
000
aaaa
aaa
aa
a

Triangular Matrices
 Theorem: (basic properties of triangular matrices)
 The transpose of a lower triangular matrix is upper
triangular, and the transpose of an upper triangular
matrix is lower triangular.
 The product of lower triangular matrices is lower
triangular, and the product of upper triangular matrices
is upper triangular.
 A triangular matrix is invertible if and only its diagonal
entries are all nonzero.
 The inverse of an invertible lower triangular matrix is
lower triangular, and the inverse of an invertible upper
triangular matrix is upper triangular.

Triangular Matrices
 Example:
 The matrix A is invertible, since its diagonal entries are
nonzero, but the matrix B is not.
 This inverse is upper triangular.
 This product is upper triangular.










−
−
=









 −
=
100
100
223
,
500
420
131
BA









 −−
=










−
−
=−
500
200
223
,
00
0
1
5
1
5
2
2
1
5
7
2
3
1
ABA

Symmetric Matrices
 A square matrix A is called symmetric if A = AT
.
 A matrix A = [aij] is symmetric if and only if aij=aji for all
values of i and j.






















−





−
−
4
3
2
1
000
000
000
000
,
705
034
541
,
53
37
d
d
d
d










−
705
034
541

Symmetric Matrices
 Theorem: If A and B are symmetric matrices
with the same size, and if k is any scalar,
then
 AT
is symmetric
 A+B and A-B are symmetric
 kA is symmetric
 Theorem:
 If A is an invertible matrix, then A-1
is
symmetric.
 If A is an invertible matrix, then AAT
and AT
A
are also invertible.

Exercise
 Find all values of a, b, and c for which A is
symmetric.
 Find all values of a and b for which A and B are both
not invertible.










−
+
+++−
=
720
53
2222
ca
cbacba
A






−−
=




 −+
=
7320
05
,
30
01
ba
B
ba
A

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