(vii)

4°4. Use ofthe Inverse of a Matrix to Solve Simultaneous

Linear Equations 131
Rank of a Matrix
51. Introduction
a Matrix
139--113992
52'1.ofLinear
S2. Rank Dependence and Linear Independence 139
of Vectors
Elementary Operation or Elementary Transformation 140
5°3. Rank
of a Matrix and its
5:31. Elementaryof Operations or Elementary Trans. 146
formations a Matrix 146
532. Elementary Matrices
53:3. Equivalent Matrices 147
153
53-4. Reduction of aMatrix to Triangular Form
Elementary Transformations and Rank of a 154
53'5.
Matrix 156
536. Use of Elementary Transformations in Com
puting the Inverse of aNon-singular Matrix 184
6. Linear Equations 195-218
61. Introduction 195
62. Homogeneous Linear Equations 196
6:3. Non-Homogeneous Linear Equations 205
7. The Characteristic Roots and the Characteristic Equation
of a Matrix 219-244
71. Introduction 219
711. Nature of Characteristic Roots of Special
Types of Matrices 226
72. Characteristic Vector 230
73. The Cayley-Hamilton Theorem 236
8. Quadratic Forms 245--283
81. Bilinear Forms 245
82. Quadratic Form 246
8:3. Linear Transformation 249
831. Linear Transformation of the given
Quadratic Form 249
8:32. Efect of a series of Linear Transformations
on a Quadratic Form 250
833. Invariance of Quadratic Form under Non
singular Linear Transformations 251
8:34. Invariance of the Rank of Quadratic Form
under Non-singular Linear 252
84. Reduction of Quadratic Form intoTransformation
Sum of Squares
(Canonica! Form) 252
841. Analytical method of Reduction of Quadratic
Form to the Canonical Form 257
85. Index and Signature of Real Quadratic Form 260
86. Definite Real Quadratic Forms 265
Bibliography
Index
284
285-287
 CHAPTER ONB

ALGEBRA OF MATRICES
11. NoTATIONS
in n
Consider a system of m simultaneous linear equations
variables X, *?, ..,X given by
a11*1+a12xt...+a1nx,=b1
a21X1+az2X,t...taznXn=b ..")

am11t ae2t...+amntn=bm
then
If we think of each linear form in () as a single entity,
) can be expressed as a single equation as follows : b1
a1x1ta2x2t...ann
b2
ag1X1+az2X2+...taznn

am1X1+Qm2N2t.tamnn bm/
Bquations () can be more conveniently written as
AX=B
where
b1

U22 ... a2n

be
W21 B=
A=
X=

am1 am2 9
bm
linear form (*)
A is known as the matrix of the
Definition. The system :
a11

a21 a22 ... @2n

...(1'1)
A=

Qml

of 1nelements of a field F arranged in the

form of an ordered set
IM-S.78.1
2 AN INTRODUCTION TO MATRICDe
of mrows and n columns is Known as mX, (read as mby n)
over F matrix
For example,
10 1

() A =
23 5 is a 3x3 natrix over N.
13 0
V2 1

-1 2 is a 3X2matrix over R.
(ii) A =
V3

More precisely the matrix (11) may bematrix written as A=(a)

Where ais are called the elements of the ; the suffixes
columns
i(el,2,... , m) and j(=1, 2, ... , n) refer to rows and respec
tively.
Remarks 1. In general, the matrix may be defined over any
set S.
2. It should be borne in mind that a matrix is an entity by
itself and should not be confused with a determinant. A determinant
numerical
has a single numericai value whereas a matrix has no numbers
value but is only a convenient way of expressing system of
onwhich the operations of addition, multiplication, etc., are defined.
12. VARIOUS TYPES OF MATRICES
Row and Column Matrices. In (1·1) if we take m=l, we get
1xn matrix
A=(a1:, a,2, ..., a1,) ...(12)
which is called row matrix or row vector. Similarly on taking n=l
we get m X1matrix
l11

A = ...(120)

which is cal<ed column matrix or colunn vector.

Zerefor Null Matrix. Any mXn matrix with allthe elements
as zero is cailed a zero or null matrix. It is usually denoted by
Omzn or sirnply by O. Thus, for example
/0 0 0
ALGEBRA OF MATRICES 3

Saáre Matrix. A matrix in which the number of rows is

equal tom=n,
in (1:1) the number
then of columns is called a square matrix. Thus if
A=(a), (i, j=1, 2, ... , n)
is called nXn square matrix or a square matrix of order n.
Diagonal Matrix. In a square matrix, the elements au, (/=1,
2, ..., n) are known as diagonal elements and the line along which
they lie is known as the principal diagonal or leading
diagonal.
A square matrix in which all the elements except
principal diagonal are zero, is called a diagonal matrix. those in the
Thus the
matrix

A=(a;) ;ij= 1, 2,.., n)

is a diagonal matrix if a_j=0, i#j.
A djagonal matrix of order n with
an: is denoted by diagonal elemeits a11, az2, ... ,
A=diag (a11, a2g, on, ann) ..(1"3)
Remark. The sum of the elements in the principal diagonal
of a square matrix A is called the trace of thenatrix and is abbre
viated as tr(A). Thus if A=(a,;) is asquare matrix of order n, then

tr(A)=
.. (13a)
i=1
Scalar Matrix. A diagonal matrix whose all the
elements are equal is called the scalar matrix. Thus thediagonal
square
matrix A=(a) is a scalar matrix if and only if for some
scalar k,
aij=k,
=0, i=J;,j=1, 2,", n
and we write
A=diag (k, k, ..., k) ...(14)
Unit Matrix or ldentity Matrix. In (1'4) if we
A is called a unit matrix or an take k=l, then
denoted by In. Thus, identityy matrix of order n and is
I,=diag (1, I,..., 1) ..(15)
For example,
l 0 0

are unit
I=
() and I3=

matrices of order 2 and 3 respectively.

\0
0 1
0

Remark. An identity matrix may also be defined as follows :

 4 ÀN INTRODUCTION TO MATRICES

The matrix A=(o,) is an identity matrix of order n if

By=1, ij)
=0, i#)
j=l, 2,.,,
Öuy, So defined is known as Kronecker's delta.
8mmetric and Skew-Symmetric Matrices. A square marik
A=(ay) is called symmetric if
aiy=a for all i, j t ...(1'6)
and it is called skewsymmetric if
ay=-a for all i, j .(1'6a)
If i=jin (1'6a), we get
a=-4is i=1, 2,.., n
ay=0, + i=1, 2,..., n
Thus every diagonal element of a skew symmetric matrix is
necessarily zero.
Triangular Matrix. mXn matrix A=(au) is said to be upper
triangular, if
ay=0, i>j
and it is said to be lower triangular if
ay=0, i<j.
Thus if A is a square matrix then in an upper triangular
matrix all the elements below the principal diagonal are zero while
in lower triangular matrix all the elements above the principal dia
gonal are zero.
Remarks. 1. Triangular matrix need not be square.
2. A square matrix which is both upper triangular and lower
triangular is a diagonal matrix.
Sub-Matrix. The matrix obtained on deleting any number of
rows and columns of the given matrix A is called the sub-matrix of
A. For example, if
1 1 -1 5
A= 2 -3 4 6

3-2 3 7
is a 3x4 matrix then the matrix
/2 -3 4
B=
\3-2 3
obtained on deleting the first row andthe last
2x3 sub-matrix of A. column of Ais a
"said toCouparable Matrices. Two matrices A=(ay) and B=(by) are
be comparable if each has the same
columns as the other, i.e, if they have the same
number of rows and
dimensions.
ALGEBRA OF MATRICES

EAuality of Matrices, Two matrices A=(ay) and B=(by) are

sajdto be equalif and only if
() they are comparable, i.e., they are of the same dimensions
and
(i) the elements in the corresponding positions of the two
matrices are same, .e., for each pair of subscripts i and j we have
aiy=biy
Renark. The equality relation in the set of all matrices is an
enuivalence relation since the following three properties (for equiva
lence relation) can be easily verified.
()If Ais any matrix then A=A (Reflexivity)
(ii) If A=B then B=A (Symmetry)
(tii)) If A=Band B=Cthen A-C (Transitivity)
1:3. OPERATIO NS ON MATRICES
Addition of Matrices. Two matrices A=(a) and B=(b) are
said to be conformable for addition if they are comparable and then
their sum A+B is deined as the matrix
CeA+B=(c) ...(17)
where
Ciy=aytby ...(17)
ie,the sum of twO matrices is obtained on adding their correspond
ing elements. Obviously A+B has the same dimensions as A or B.
For example, if
11 a13 b11 b12 b1s \
a21 and B= b¡1 bn bes
bi b¡2 bg3
then
d11+b1 1tb1s 1s+b18
A+B= as1+ba1 a2tba azs+bzs
as1+bsi ds2-+bss ss+bss
131, Lawsof Matrix Addition.
If A=(ay) B=(b) and C=(Ci) are three matrices each of the
same dimensions mXn, then-the following laws of algebra hold.
(1) Commutative Law
Theorem 11; A+BeB+A ...(1"8)
Proof. We have
A+B=(aj+bu)
-(biy+au)(By associative law of numbers)
=B+A.
 AM MIBONnON 19 MAB
(2) Aoeiative Law
Theorem 12. A+(B4C-(A +B)4C
Proof. Since A, IB And C Are
order mxn, the matrices A, () +C), (A comparable matrices eh t
+5) And Care also of sae
dimenslons mn and thus the matrix additions A (B C) 4nd
(A4B)+C are delined and these matrices are comparahle, Funthey
(, J olement of A+(B+C)
-a+butol
-\ay tbyl+ ca (By aSsOciative law of numbersy
-(, j element of (A +B) +C
Henco the result.
(3) Dlstrlbutve Law
Theorem 13. k(A +B)-kA +kB
k belng an arbltrary scalar ...010)
Proof. We shallSirst definc multiplication of a matrix by a
8calar. If A=(a) is m%nmatrix the and k is any scalar number, real
or complex, then the product of k and Awritten as kA is defined as
kaj1 ka,2...kais
kai kazz kan
kA -(ka,) ...(111)
kami kant...kamn
Obviously. we have
kA Ak
i.e.,the matrix product with a scalar obeys ..(1110)
commutative law.
Matrices on both sides of (110) are comparable.
(6, J) clement of k(A+B) By definition

-k{a4+b]=kay+kby
=kx(1, J) element of A
+kx(1, element of B.
Hence the result.
Remark. In (111)if we take k=-1, we get
-A=(-a) ...(111b)
which is called additive inverse or negative of A.
13-2. Some More Properties of Matrix Addition
(1) Existence of Identity. IfA is mxn matrix and O is a null
matrix of the same dimensions then
A+0=0+A=A ...(112)
 AN
8

that
Let A(ay)conformable
they are
matrix
be mXn for
the and IproductNTB=(b)
ROD.UCTION "TO MATNC
be
product
CAB-(c) "Then he
is a mXp matrix where cy is
the
inner
mat z
the j column of B. Diagrammatically, product of th
row of Aby
C91

C91

Cml Cm2-.Cmg.Cmp

b1l.. b! ...b1,
bs1... b ...ba,
. .bip
bay
Vhere

bss

...{1.16a)

In other Words, (, j element of the product AB is

obtained ou
multiplying the elements in the jth row of A with the corresponamy
elements inthe j column of B and adding the resulting produsl
Remarks. 1. If Aand Bare Square matrices of the same order,
say, n then both the products AB and BA are defined and each is a
square matrix of order n.
2. In the matrix pfòduct AB, the matrix Ais calledthe pre
multiplier (pre-factor) and Bis called post-multiplier (postfactor).
3. The sufix k used in the summation formula (116a)is
result.
called the dummy sufix which does not inthe final
It can be replaced by any other letter exceptappear
iand J.
 ALGEBRA OF MATRICES 9

Example 11. If
-2

A-( B
and B= 3

-1
0

find AB and BA.

Solution. Here A is a 2x3 matrix and B is 3x2matrix so
that both the matrix products AB and BA are defined, AB being
2x2matrix and BA being 3x3 matrix.
1 -2
2 3 1
AB= 0
-1
2

2x1+3x3+1 x(-1) 2x(-2)+3 x0+1 x2

-

0x1+2x3+(-1)x(-1) 0x(-2)+2x0+(-1)x2
10 -2

-2

BA=

-1
3
)
1x2+(-2) x0 1x3+(-2) x2 1x1+(-2)x(-1)\
3x2+0x0 3x3+0x2 3x1+0x(-1)
\(-)x2+2x0 (-1)x342x2 (-)x1+2x(-1) /
2 -1 3
6

-2 1 -3
Properties of Matrix Multiplication
Property 1. Matrix Multiplication is, in general, not commuta
tive.
Theorem 14. For two matrices A and B, AB#BA, in general.
Proof. If the matrix product AB is
that the product BA is also defined. For defined,
example,
it is not necessary
A and n XpmatrixB, the product AB is defined butfor the
m Xn matrix
is not defined since the product BA
number of columns of B is not equal to the
number of rows of A. Moreover, even if both the products AB
and BA are defined they need not be equal. For
example, if
 l0 AN INTRODUCTION TO
MATRICES
-()
A=
and B=

$o that both AB and BA are defined, we have

ba
AB= and BA=
-a' -ab/
AB4BA
Another numerical illustration is given in Example (11).
These examples show that matrix multiplication is, in general.
not commutative. There is, however, an imporlantexception as given
below.
If A is a square matrix of order n and I, is an identity matrix
of order nthen
Ih.A=A. h=A ..(117)
In general every square matrix of order n commutes with every
Scalar matrix of order n.
Property 2. Matrix Multiplication is Associative.
Theorem 15. If A, B and C are matrices of suitable dimen
sions so that the matrix products A(BC) and (AB)C are defined then
A(BC)=(AB)C ..(113)
Proof. Let A=(44), B=(b) and C=(C) be mxn, nxp and
IM-
M.
pXq matrices respectively so that AB=(e) and BC=(Vy) are mXp
and nxq matrices respectively where

...)
k=l

end .(**)
bigcg
Thus (AB)C=(w),say, and A(BC)=(H), say, are each mxq
matrices where

Uir Cry and x= ...(***)

0M,
r=1 k=1

To establish (1:18) we have to prove that wy=X. Now

Wiy= [From ())

 iM.
IM EM
ALGEBRA OF MATRICEs

|From *l

Hence the result.

Remark. In view of the above result, we may use a
symbol ABC to denote either A(BC) or (AB)C so that single
A(BC)=(AB)C=ABC ...(118a)
Corollary 1. Generalisation of the Associative Law. By
mathematical induction, the
to the case of n matrices A1, above result can be easily generalised
A, .., A, so that the product of any
number of matrices is independent of any particular
bracketing, provided it is meaningful. Obviously the order manner of
matrix factors cannot be changed. Thus, for of the
matrices A1, A2, A3 and A4 we have example, ior four

A,(A{AgA4))=(A1(AgAs4=A1A)(AsA4)
-((A1A)A;)Aq=A1AzAsA4 ....(1186)
provided the products are define.l.
Corollary 2. Positive Integral
For any matrix A, the product AA is Powers
defined
of a Square Motrix.
only if Ais a square,
natrix. We write AA=A?, By associative law of
we have multiplcation,
A2.A=(AA)A =A(AA)= A.A2
Thus A? multiplied by A or A multiplied by A? gives the same
which we denote by A. Thus result
A.A=A.A'=AAA=A3
Hence, in general, the product
A.A....A (m factors)
is independent of any particular manner of
cation (c.f. Cor. 1) and is denoted by A", bracketing for multipli
Definition. If A is a square matrix of order nthen
A'=A
and Ak+l=A*.A
for each positive integral k.
} ...(119)

Theorem 16. If A is a square matrix of order nthen for

positive integral values of p and q,
(i) AP.Ao AP+0 ...() ..(19a)
(i) (AP)o=AM ...(*)
12 AN INTRODUCTION TO MATRICFR

Proof. () By defnition (1'19), we have

A9.Al=A41
Thus () is true for q=l and for all values ofp. The result
will be established by mathematical induction. Let (^) be true for
a fixed value of q=r (say) and all values ofp. Then we have
AP.A'=APtr
Also AP.AP+l=AP,(A".A) [By def. (1'19)
=(AP.A)A [By associative law)
-(AP+) A [From (*)
=A+n1 (By def. (1'19)]
-AM) [By associative law for
addition of numbers]
Thut if () holds for q==r and all values of p, it also holds for
q t land all values of p. The result now follows by mathematical
induction.
i) (A)=A.AP...AP (g times)
(From part (i))
AM,
Property 3. Matrix multiplication is distributive with respect
to áddition of matrices.
Theorem 1%. (Distributive law)
A(B+C)=AB-+AC
and (B+C)D=BD+CD .(120).
provided the matrices A, B, C and D are conformable for the opera-.
tions involved.
Proof. Let A=(au) bem Xn matrix and B-(bu) and C=(ey)
be cach nxp matrices, so that (B+C) is also nXp matrix. Thus
A(B+C) is of order mxp. Also cach of the matrices AB and AC
are of order mxp and so is AB+AC. Thus the matrices A(B+C)
and AB-+ACare comparable. Further
(i,j) element of AB+C)

k=1
(Since in case of real numbers, multipli-.
cation is distributive w.r.t. addition).
-(,j element of AB+U, j) element of AC
=(t, j)elemer:t of AB+AC.
 AN
14 INTRODUCTI
values of m.
ON TO
MATRICES
integral
We are given that for
a~ß(2m+ 1)n/2
cos (a~B)==cos {(2m +1)/2)=)
Substituting in (°), we get
AB-=

x 2.
a null matrix of order 2
Example 1l13. IfA and B be matrices such that both AB
A+B are defined then provethat both A and Bare and
of the same order. square matrices,
Solution. Since the matrices A and B are
addition, they must be of the same dimensions.
matrix, then B is also mXnmatrix.
conformable
Thus if Ais for
mxn
Further, since the product AB is defined
columns of A must be equal to the number of the of number
mnst have n=m and therefore both A and B are rows of B, i.e., we
the same order n. square matrices of
Examole 14. Prove that the only natrices which are
tative for multiplication with a diagonal commu
gonal eiements are diagonal matrices. matrix with distinct dia
Solution. Let A be a diagonal matrix of
elements, i.e., ordern with distinct
@11 0 ...0

a22 ...0
A=

0
such that
autas, (i#j).
Let B=(bi) be any matrix. In
B, i.e., AB=BA, B must be a square order that A commutes witn
matrix of order n. Nw
AB=BA
if and oniy it

by (au-as)=0
Thusb=0 when
elemerts of B are auta, i.e., when i#j. Hence all the non-diagonal
zero and
therefore
Example 15. If Adiag (a, b, c),
B is a diagonal matrix.
show that
A"=diag (a", b", cm) ...()
ALGEBRA OF MATRICES 15,

Solution. We havc
a 0

0 0
A=diag (o, b, c)=
C

00 a 0 0

A2= 0 b 0 0 b 0

00 C 0 0 c
a2 0 0

diag (a2, b2, c2)

00 c?
Thus () is true for m=2. Let (*) be true for m=r, (say) i.e.,.
A'=diag (a, b', c') ...**)
Then
a' 0 0 a 0 0

A+l=A.A= 0 b 0 0 b 0
0 0 c 0 0 c/

br+1 0 =diag (a+1, br+1, c+1)

Hence if () is true for m=r, it is also true for m=r+1.

But we have proved that () is true for m=2. Hence, the result
follows bymathematical induction.

Example 1'6. If A= prove that

(al+bA)"=a"I+na-1 bA
where I is the 2 rowed unit matrix, n is a positive integer and a, b
are arbitrary scalars.
Solution. We have,
A=
()A3=A2.A=0, A4=A3.A=0 and so on.
we
Also since A is a 2x2 matrix and I is also a 2 x2 matrix,
get
...()
Al=[A=A
 AN INTRODUCTION To
Also we bave
MATRICE
I=['=P.=·
Bybinomial expansion [ (c.f. Q 17(iv) in cxercise setI(D 1
(al+bA)P=(a+() (a-,bA,
otber terms being neglected since A, A,...A" are all null matrices
Hence on using (*) and (**) we get
(al+bA)^= I+na-, bA,
as desired.
Example 17. Explain, why in general
(a) A-B²(A +B) (A-B) (Nagpur Univ. B.Se. 1973,
[Saugar Univ. B.Se. 197h
(6) (A+B,²$A42AB+B ? (Saugar Univ. B.Sc. 1965)
Solution. (a) By ditributive law, we have
(A+B) (A-B)=(A+B) A-(A+B) B
=A+ BA-AB-B4,
Since, in general, AB4BA, we get
(A+B) (A-B)A-B,
However, if Aand Bcommute, i.e., if AB-BA, then we get
(A+B) (A-B)=A-B!
(6) (A+B)=(A+B) (A+B)
=(A+B) A+(A+B) B (By distributive law]
A+BA+AB+ B'.
Since, general, AB+BA,
in
(A+B)'#A+B®42AB, in general.
However, if A and Bcommute, i.e., if AB=BA,
then
(A+B)²=A+2AB+B,
Similarly, it can be proved that
(A-B)=A-2AB+B,
provided A and B commute, i.e., if AB=BA.
Si Example 18. A is a square matriz of order n with all
equal to unity and B is a matrix of order a with all elements
elements equal to n andsquareother clements n-r. Shew that diagonat
Deduce that A2nA.
(B-r) [B-(r-nr+r) I]=0.
Solution. Since ¢is. nXn square matrix with
all the
unity, we get
1 1...1 1 1...1
eements
n...n
A= 1 1...1 1 1...1
n...n

1 1...1 1 1...1 n n...n

A2-,nA=0 A (A-nl)=0
*-()
 ALQRIRA OF MATRICES 17

Also
/n-tr n-to . n-to\
Bam n-rto n-r+r... n-r+o

-+o n-to ... n-r+r/

=(1-) A+rI ...")

A (B-rI) (B-(n-nr+r))
=(n-r)A [(n-r)A+r-(n-nrtr)I)
[From (*) 1
-(1-r)A.(7-r) [A-n
-(a-r)A [A-I]
=0 [From (*)]
Example 19, Show that the possible square roots of the two
rowed unit matrix I are
Probu
+I and y where 1-al=PY.

Solution. If the matrix

(;) is a possible square root

of the 2 x2 unit matrix I, then we have

.

(;)(;
a2+BY B(«+8)
( Y(«+8) yB+82 H): )
a?+By=1l3+82
and P(a+8)=0=y(a+8) . (*)
Now () gives (0o")
Case (i). If a=-8 a + 0then(*)is automatically satisfied
and ()gives a4By=1, Thus in this case the square robt matrix
becomes
IM--578-2
 18 AN
INTRODUCTION TO
where 1-a2=By
MATRICES
Case (ii) If a=8then (**) > =0=y and hence
al=82=1L> =8=+1. from,(") we get
yato Hence in this case, the square root matrix is given by
t1 0
ie, i.e., tI
Rachal 0
shatn Hence the result.
EXERCISE 1(1)
1 4 -1 -2
1. If A= 3 2 and B=| 0
2 5 3 1
find the matrix C such that
A+B-Ca0, i.e., null matrix.
2. Is itpossible to define the matrix A+B when
(a) A has 2 roWs and B has 3
rows.
6) A has 3 columns and Bhas 4
Kc) A has 2 rows and B has 3 columns.
( Both A and B columns.
are square matrices of
Give reasons. the same order ?
1 2

-1 0
and B
find 2A-3B.
(6) Compute 2A+B-C when
A= 3 4
B=

Yc) Show that matrix

addition of matrices. multiplication
Univ. B Sc is
[Delhi
B.Sc. distributive NanakW.r.t. the
1974; Guru
1975(S); Poona
Agra Univ. B.Sc. Univ. Univ.
B.Sc. 1970;
4.
B.Sc. 1969;
Is it possible to define the matrix Gorakhpur1970;Univ.Banaras
B.Sc. Univ.
(a) A has 2 rows and B has 3 rows. product AB 1968
(b A has 2 columns and B has 4 columns.
when
(cX A has 3rows and B has 3 columns ?
 ALGEBRA OF MATRICES 19

5. (a) Ir show that

AB= BA.
2 3
1 -2 31
and B= 4 5
2 5/
2 1
find AB, BA, and show that ABBA.
6. (oYShow by means of an example that the cquation
AB=0, oes not necessarily imply that at least one of the matrices
A and B must be a zerO matrix. (Delhi Univ. B.Sc., (Hons.)1973]
6) Show by means of an example that it is possible to» have
matrices Aand B such that AB=0 while neither A=0 nor B=0.
[Agra Univ. B.Sc. 1971; Delhi Univ. B.Sc. 1970;
Kanpur Univ. B.Sc. 1970;Poona Univ. B. Sc. 1970;
Gorakhpur Univ. B.Sc. 1967]
(e) Is the following statement true ? Give reasons in favour
of your answer :
If A, Bare n-rowed square matrices then
AB=0 ’ at least one of Aand Bis zero.
[Gujarat Univ. B.Sc. 1970]
(d) Give an example of matrices A and B for each of the
following:
AB is defined but rot BA
Ai) AB=0but BA#0 [Bombay Univ. B.Sc1968].
Taking
00 0 231
A= 0 0 0 B= 5 0 1 6
OELH

|1 1 1 -4 2 1 -4
verify that, if AB=ACand A#0, it does not necessarily fo'lcw that
B=C.
COS a -Sin cos B - sin B
8. If A B=
sin a COS a
sin
show that AB=BA.
9. (a) Show that matrix multiplication is associative, if con
formability is assured, i.e.
20 AN INTRODUCTION TO MATRICES

A(BC)=(AB)C,
if A, B, Care mXn, nxp and p Xqmatrices respectively.
[Delhi Univ. B.Sc. 1973: Meerut Univ. B.SC. 1972;
Poona Univ. B. Sc. 1970: Ranchi Univ. B.Sc. 1969;
Allahabad Univ. B.Sc. 1967)
Ab) Compute the products A(BC) and (AB)C, given that

C=(1,-2)
B=| -1

and verify the associative law of multiplication.

10. If A be any mXnmatrix, then prove that
Ln A =A=Al,
11. á) Given that

(;)CJA.
determine x, y in terms of a, b.
(b) If
3 4 5
(8x+3y 6z 32

(: -1 0 -2
3 4
4 12 26x--Sy/

find the values of x, y and z. [Kolhapur Uniy. B.Sc. 1972]

will
(c) For what value of x,
2 1 0
(* 4 1) 1 02
02 4 J
n), (2, m, Ag) be the direction
12. If (l1, my, prove thatthe matrix product cosines of
perpendicular lines twO
hm1 lang
lgmg mg?
hm, m? men2
n mana n
[Calcutta Univ,
B.Sc.
0S a
zero
matrix.
(Hons.) 1969)
ALGEBRA OF MATRICESs
21
Hint Use :

h4+mm,tning=0 ,3+mj2+nj1
Istmms + ngng=0 and +mg+n,2=1
hbtmmg +nng=0 l32-+m34ng2=1.
13. If A= show that A?0.

Remarks. Thisexample shows that

A=0 A=0
14. (a show that

1+2k -4k
show that
k 1-2k/
[Kanpur Univ. B.Sc. 19701]
cosh a sinh a
(c) If A= prove that
sinh a cosh a
cosh na sinh n.
sinh na cosh na
Hint. (a)(b) and (c): Use mathematical induction.
[Allahabad Univ. B.Sc. 1970]
Kd) Let f()=-5x+6, find f(A) if
2 0

A= 2 1 3 (Delhi Univ. B.Sc. 1970]

1 -1 0 SRI
1 2 2
VENKATESWAA LULLEGE, LIBR ARY
NEW DELHI-|I0021
2 1 2
shÑsftbaë344r0
2 2
A24A-SI=0
[Meerut Univ. B.Sc. 1973]
-2 3

2 -4
3 -5
22 AN INTRODUCTION TO MATRICER
find the value of the matrix
3A-2A+5I where I is a unit matrix of order 3.
(Delhi Univ. B.Sc. (Hons.) 1970)
cOs 0 sin
18. If Ag
\-sin & COS
then show that
cos no sin n
-sin n cOS no
TOT every positive integer n, Further show tbat A. and Ag commute
and that
AAg=Aa+8.
[Rajasthan Univ. B.Sc. 1973: Ravi Shankar Univ. B.Sc. 1970:
Meerut Univ. B Sc. 1969)
0 1 0 0
16. If A=
and B-=
0 1
calculate the matrix products AB, BA and
Show also that if n is a scalar then A2B+BA.

17,
(A+nB)=nl
If AandB are n-square matrices which commute and
is a positive integer then prove that
(i) (AB)'=A22AB+B²
(i) (A+B)(A-B)=A?B2
(ii) (A+B)°=A343AB-+3AB2+B3
(iv) (A+B)'=A'+()A-.B+()A-2B²+...+B
() (AB)=A'. B.
Hint (iv) and (v)., Use mathematical induction,

fA , choose a, Bso that

(«I+BA)P=A.
19. If show that
(al+bE)=I+34bE,
aand bbeing scalar. [Gorakhpur Univ.
20. IfB, Care n-square matrices and if A-B+C, tB.henSe. 1970)
AP+i=B[B+(p+1)C),
is a positive
provided Band C commute, C2=0 and p integer.
ALGERRA OF MATRICES 23

Hint. We have
A'=(B+C)=(B+C)(B+C)
-B+2BC; ('.: BC=CB, C2=0)
=B(B+2C)
Thus () is true for p=l. Now establish the result by mathe
matical induction.

21. Classify the follcwing statements as true or false, giving

reasons. Also correct them if they are false.
(i) Anull matrix is always a square matrix.
(ii) A matrix in which all the diagonal elements are equal, is
called a scalar matrix.
(iii)) Two matrices are said to be equal if they have the same
number of rows and the same number of columns.
(iv) If a is scalar and A is any n Xn matrix then aA is a scalar
matrix.
(1) A matrix with each element as unity, is called a unit
matrix.
(vi) A matrix A=(a,)is said to be triangular if ay=0 i j .
(vii) If A and B are matrices such that AB=0, then either
A=0 or B=0or both are zero.
(vii) IfA and B are square matrices cach of order n then
(A+B)=A't2AB+B²
and (A+B)(A-B)=A2-B2
(ix) IfLis a unit matrix of order n and A is a square matrix
of order n then
A+h=A.
22. (a) If A, B and C are three matrices such that
a h g

A=(* y z), B= h bf C=
2

prove that
ABC=ax+by2+ cz2+ 2hxy+2gzx+2fz
(Gorakhpur Univ. B. Sc. 1964: Agra Univ. B.Se. 1972 :
Nagpur Univ. B.Sc. 1973 ,: Ranchi Univ. B. Se. 1970]

(b) If

prove that (A-21)(A-3I)=0

[Burdwan Univ. B.Sc. 1970]
 ALGEBRA
OF

24
AN INTRODUCTION TO MATRICES
6. (C)

23. (a) If A= and find (b

3 9.
AB and BA. Is AB-BA ? [C.A. (Intermediate) May 1975} 11. (a
(6) If and show that

A2-(a+d)A=(bc-ad)I.
(I.C.W.A. (Final) June 1975
(c) The matrix A is given by 14.
1 2

-3
undthe matrix2. for the polynomial A-+3A+5I where I is1974
matrix of order unit 16.
[I.C.W.A. (Final) June
-1
24. If A=
2 B= 18.
and
(A+B)2=A2+B2, find a and b.
25 (2. Prove that any pair of diagonal matrices of order 2 X2 23.
commute,
(b) Find the matrices A and B which satisfy the following tWO 24.
conditions simultaneously:
i) AB=0 but A#0 and B40 25.
(i)A+B=2I but A#I and B#I.
ANSWERS 1()
0 2

1, C= 3 7
6

2. (a) No. (b) No. (c) Yes, provided Ahas 3 columns and n
has twO rows. (d) Always.
-4 -12 3
3.
-14 -9
(6)

4, (a) Yes, provided A has three columns.

-( 2 -9

(b) Yes, provided B has two rows.

columns.
(c) Yes, provided A has three
6

AB-BA-{
5. (a) AB=BA: 0 12
-10 2
-4 21
BA= -16 2
(b) AB 10 3 -2
-2
37
11
ALGEBRA OF MATRICES 25.

6. (c) No.
-3
9. (b) A(BC)=
-14/
11. (a) *=1-2a, y(2-b)
(b) x=1, y=3, z=4 (c) x=-2ti6
-1 -3 21 -17 1:
14. ()-1 -1 -10 13 34 -2
-s 4
22 25
1

16. A2.B+B².A= 0

18. Either a=ß=A

V2 Or
a=-p=t
3
23. (a) No. (c)
\-12 -1
24. a=1, b==4
2
25. A=
B=