Linear Algebra

ptro ductio n
~ere arc: tw(J typcH qf lin ear 1-;imultan cow-~eq uations,
(i) AX /J ( N ,111- homogcr ww, lin ear cyuati om,J.
(ii) AX (J ( lf()tr,ogc nou H lin ear cquation 8).
Thtrc an; varnJui1mcth,,d11 UJ Holvc the ahovc cyuatiun s as substitution, eliminat ion, cross-mu ltiplicati on
etc, Solvi ng th c l inear r1y f{ lc m cyuati on by matrix rnc1hod is caJJed Linear Algebra. /
The w<m-1 ,natr'i,~ wa~ introtJuccd in l ~~-,() by a French Mathematician \cayJ~ 1

1>efin ition of Matrix

Matrix i il th e eul k ction ,,r numbcn~ (real or compkx ) in fixed number of rows and columns . The
h'Of'Jliontal :trr:1 y H of' a rrwtrix arc calk<.l ROWS and the vertical arrays are ca])ed C OLUMN
S.
A Hct 1Jf' m,, n riumhcrn (real ur c;omplcx ) arranged in the form of rectangu lar array having
m rows
and n rn lun11rn lhctt m / n iri rnll c<.J order of th<.: matrix .
u,, u, ,. a,n
(,/,J I u 'J;;, (j,J, n
A

{),11,I u1,,, ljll/11 ,. ln, /fl

The 11 lu1 v1; rn:1t rix ~;11J be rcprcHent.cd in <.: ompac.;t form a8,

A I a,, I"'""
'~,, ·1 1m1td x all iJhown below,
( '0 )111111 11;

( ' (' ( '

I J \
Mntrix[NumeJ Elemelit
1 1 1 ,1, J.
(I ll 11 11 ,,, ., I(, \ [A] = {a,1} R -,,.c
II 11 1, f1011ltlon io row
'' JJ 11 11 I ., /(1 J!<own --1J t L Number of columns
11 11
Po"ltlort in column - ·· Number of rows
11 11 (I H 'I- /( i

or 11 111111, ix, 11 iH 11,;cc,wury 111 dc f111 c it ,; orde r 11nd ilHc lcrnonlH.

C.pyrt1ttt ll fllHI 1 m1 1,1 u I i\/11 ,1 ll l'i , 1'4 1111111 N1tMlll', llhll1tl w.u.,. C 1111h11JI I 9 7 1:1 11 ,ll~f•, •}~H4J/l'l•ll7'1 www.pC.ac:t1 •,ftlY•CO·!~
11,.,,.,, ,,,,,,,. , ,,.,,,.,,, .,_-. , ·1•11,u .•111111·1, 111..,,,.,, ~, , H•1~•11 .11 ·111~,. ' '"'"' "· -., , 11·11.M' !.'.u www.1'1ectboolJ,com/-tNC"4el'rlV ,
 GATE ACADEMY ~
1.2 EnglnHrlng M1them1tlc1

Basi c Oper ation s of Matr ix

1. Tran spose of a Matr ix:
hy intl.:rchanging the rnws with the
The trnnspose of a matrix A written as ;(" (or A'), is obtuined
corresponding columns of A.

a b c ll d]
Example: lf A=[d e .]. then A.,, = b e
,/ . M / '
C
r - · 1, 2

2. Deter mina nts of Matr ix:

cxpc1ni,;ion or value of the matrix
Determinant is only valid for sq uare matrix. Determinant is the
ient = (- 1/' 1 .
according to the elements position coefficient. The position coeffic
01 2
=[a,, ]
Let us consider matrix of order 2x2, A
-a2, an 2, 2

2
a, , =a, ,a22 - u, 2a2,
Then, the determinant of given matrix is IAj =/a,' ti22 2,2
. . Jll21

2 3 2
ll22 Cl23 I I Cl21
· a, a, j
and matnx of order 3x), A= aa,, a a then /A/ =a,
21 22
I 23 1
· - a, 2
Cl3 I
a23 I + al) ,a_, an'
Cl33 Cl31 Cl32
C/32 Cl33
[ G31 G32 G33

expanding its row or column.

Similarly, we can calculate the determinants of higher order by
Properties of Determinants
(ii) /AB/ =/A//B/ (iii)IA"I = (IAI)" (iv) /kA/ =kn/A/
(i) IAt/= /Al
anged , the sign of the value of the
(v) If two rows (or two columns) of a determinant arc interch
determinant changes.
the value of the determinant is zero.
(vi) If in determinant any row or column is completely zero,
al, the value of the determinant is
(vii) If two rows (or two columns) of a determinant arc identic
zero.
3. Matrix Mult iplica tion
It is valid for both square and non-square matrix , the existen
ce of resultant depends upon the order
of the matrix .
A/3 is m x p •
Let A tn><n and Bn><p are two matrices then, the order of resultant

For example : [A]= r~ ;] [ !]B] = [~

3 6 M
Jx2

] x ] +4X 3 J X 2+4X 4] [I 3
AB = 2xl.+5 x3 2x2+ 5x4 = 17 24
18]
[ 21
3xl+6 x3 3x2+ 6x4 Jx2 30 M
c::::=n
GATE ACADEMY® Linear Algebra 1.3
Here, number of elements= 6
Number of multiplications= 12
Number of addition= 6

~m~key
'l~ -:1;
:.·
Point
I. Let A 111 x11 and B,,xp are two matrices then, the resultant is ABmx p' has

(i) Number of elements = mp

(ii) Number of multiplication= (rnp)n = rnnp

(iii)Number of addition = mp(n - 1)

2. lf A is an m x n matrix and Bis an n x m matrix then
'
tr (AB)= tr (BA), tr (AB)-::;:. tr (A)· tr (B) and tr(BA)-::;:. tr(B) · tr(A)
Here, tr represents trace of matrix i.e. sum of leading diagonal elements.

4. Minor of an Element :
Minors obtained by removing just one row and one column from square matrices (first minors) are
required for calculating matrix cofactors, which in tum are useful for computing both the
determinant and inverse of square matrices. Minor of an element is denoted by M iJ.

Example : A = [ :
-4
! ~]
4 7
then one of the minors of A will be M 11 = ! I
7
= 35-4 = 31

5. Cofactor of an Element
1
The cofactor is obtained by multiplying the minor by (-1 y+ .

6. Adjoint of a Matrix
Adjoint is only valid for a square matrix. The adjoint of matrix is transpose of the cofactor matrix.

A, Ai A3 ]
then Adj (A)= B, B2 B3 where A1 represents the cofactor
Example : If A = [:: :: [
G3 b3
C, C2 C3 3x3

of element a, of matrix A.

7. Inverse of a Square Matrix

The inverse of a non-singular square matrix is given by,
-I Adj(A)
A = IAI
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 1 .4

W Key Potnt
( Il I , \ d_1 I ) ,, , HIJ .f I I

I 1111 ( ·1lJ ) /J . ,t (1 v ) ( 11 ' ) ( ;1

(\ ) A . •t = A ' ·A - I ( V I) Adj( Ad j A) A ,. I
·A
f \ 11 l .-\ dJ , I - A " · ( V Jj j) l',dj(Ad1A1 A
I I "\ ) -\ djl ·I "' l = (/Hl jA J"' ( 7.) /\rl jf fA J k.' ( /-Ji 1.A ), f. (: p
.._
Types o f Matrices According to Dimension.a {R, C)
R ows and column'-. arc all togcthcT c,ai<l to he the d imen·. irm~: <1f rriatrf:r, ~(JJ'../../): d ing tr) tr;c d :r:..e,:,.1,.1/ ,r.g
there.: a rc two type of matri x , rec tangular mat rix and '.i{Uarc rMdn x .
I. Recta ngular .\ 1atri x. :
A matrix in which the number o f row'-. i" n<Jt cq w...1 ti> the r11..tr(l t"¼'i r:,f <./J!urr.:n Vi} k:.rn,r:. a~~
rectangular matrix ( R ct:-C or m * n) ,
A - faIJ IM"n m -:t- n
2. Square Vlatrix : A matrix in which the numh{...-r r)f r<JW ':-. GI.TC c<.p.u:J tr> the; nurnr/4.-r r:,1 u./ A"?":::·--i :-i
known a s a square matrix ( R = CJ .

Types of Square Matrix

1. Diagonal "1atri x : /\ ~ua rc matr ix in ·Nhic;h all the ck-rnl--nt~~ e--r.a,-pt (£;(),.di ng d t~gr;:ia! eli.-rrll..-n~. ~t
ZCTO is known a ~ a diagonal mat rix .
r I fJ fJ

Example : A = (J 3 (J <Jr A ..:: diag (l , 3,fJJ

Lo <J f-,

W K ey Point
:v1inirnum numbe r of 1/..ern~ in a diagrmal matrir. r) f r,rck.-r n i~
A B = diag ( a,, a.r a-) / diag ( h,, h2 , h-) =diag ( a;h_, a✓,hi., aj)·:J
2. Sca lar Matrix : A diag<>nal m~trix in which all th; dh1gfJnal c;k-roc-nt ~. M(.; c;q,,1.at, ?': kn,'> ;;r~,1.-. ~ ;:.c,;J;:;r
matr ix .
3 () ()
E.x.ampk : A = 1 () 3 <, A = diag (3, 3, 3 J
\ (J (J 3 I. ,
• · . ,,. •A
- d. r,·.,l:, ,rnat r ix in which all the diagr,nal (.;ll.-rriL·r,t.~ t:1.rc; unity i·, knrnm ,;;:, vr, rt
3. u Oll , .-,atrt X : iago t1 • ._,
· 'd t ' t matr·, v The id<.,-ntity·· matr ix <)f <,rd<..,--r n j{, dl.-r1otcd by I , .
mat n;,; o r I en I y ,... .
() () I\

F, » mpl• : / ,.• \ :, () \
I () (J l j,,:

~.....-.
.,,~ ""3
_,.,,_,i,.,~¥--
&A TE ACADEMY
,S'
Unear Algebra I 1.5
l"pper Triangular ::\latrix : A square matrix A= [ a,_ is said to be
upper triangular matrix , if J
0
-= o \\-heneYer i > j.

Lower Triangular ~latri x : A square matrix A= [ a!/ ] is said to be

lower triangular matrix , if
a -= 0 wheneYer i < j .

t -._ 0 0
Examp le: .-1 = fi· _
1 ~
.J -- -. ?
__.t_ ___5_ ____ ___6:-., .
- .J.~

j- . "I'/ -
·~ ;fey Point
..,

' For diagonal and triangular matrix (upper triangular or lower triangu
lar) the determinant is equal
to product of leading diagonal elements.
Symmetric ,1amx : A square matrix is said to be symmetric,
if ~ A w·here AT or A' is
transpose of matrix A. In transpose of matrix the rows and columns are
interchanged.

Exam ple: A=[~ ! !l

3 5 6 - JXJ
..
⇒ AT =l~! !1
3 5 6 JX
, ·'~
Properties of Symm etric Matri x :
(i) If A is a square matrix then A + AT ,....a AT, AT A are symmetric matric
are skew symmetric m~trix.
~ .~ '--- - es, while .4 - AT, AT - A
----------
(ii) If A is a symm etric matrix, k any real scalar, n any integer, B square
matrix of order that of A,
then -A, kA , AT , An, A- 1, BT AB are also_symmetric matrices. All positiv
e integral power of a
_,.. - -
symmetric matrix are symmetric.
(iii)If A, Bare two symmetric matrices, then
(a) A± B, A]i±lL,1. ~re_also_s_ymmetric matrices.
(b) AB - BA is a skew symmetric matrix .
(c) AB is a symmetric matrix when AB= BA otherwise AB or BA may
not be symmetric.
3
(d) A2 A3 A4 B2 B3 B4 A2 ± B 2 , A ± B are symmetric matrices.
3
' ' ' ' , '
Skew Symm etric Matri x: A square matrix is said to be skew symm
etric matrix if Ar = - A

Exam ple: A =l~ -: =!j ⇒

3 5 0 ); )
AT =l:2-3 ~ !j =
-5 0
- A
}x3

Head Office: A/l14-ll 5, Smritl Nagar, BhHai (C.G.), Contact: 97131131

56. 958989417~
. Copyright -·--- --- - - - - 0 :9713-'•llt tl
Branch Officr : Raipur, 0 :79743-90037 • Bhopa I' ..,
ill . 8~91-1170'12,
· -
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 1 .6
GATE ACADEMY fl
l"rn p•ertk,. of ~~l '" S~ mm ctr ic
' .\htri~ :
ti ~ :, .1 , k.c" :-, mm c1 ri c ma1ri \ . the
n
1, 1 4 1:- .1 ::-~mn K\ nc m:llri :\ for II pos itiv e inte
ger .
1 HI ~- 1, .1 ~he " :-.) mm ctri c mat
ri x for 11 pos itiv e inte ger .
, u11 '-'1 1::- :1h.1.1 ~kc,\ s~, rnn
ctri c ma trix . wh ere k is a rea l sca lar.
,1 , 1 B 4B 1s aim ske w SYt mn ctri c
wh ere Bis a squ are ma tri x of ord er
.\ II po-:,. 1t1 , e 0dd inte gra l. po, \ er of tha t of A .
a ske w sym me tric ma trix are ske
e, en inte gra l pow ers of a ske w sym w sym me tric and pos itiv e
me tric ma trix are sym me tric .
If A. B arc tv, o ske v, sym me tric
ma tric es. the n
11 l A= B. AB - BA arc ske
w sym me tric ma tric es.
( 11 \ AB ~- BA is sym me tric ma trix
.
If A 1s a ske w sym me tric ma trix
and C is a col um n ma trix the n C r
4 . If A is. am squ are mat ri x
AC is a zer o ma trix . 1
ma trix .
. the n A
-
+ A r is a sym me tric ma trix and
A - A ~ ~-~ ske w sym me tric
--
W Key Poi nt ·
··
( i , The ma trix wh ich is both sym
me tric and ske w sym me tric ~s t
( ii l If A is sym me tric and Bi s ske be a nul l ma trix .
w-s ym me tric , the n tr(A B) = 0 :
( 111) An y real squ are mat
ri x A ma y be exp res sed as the sum
of a sym me tric ma trix A and a ske
sym me tric ma tri x A . 5 w
11

A= ½[A + A 1
] + ½[A - A r] = A_\· + A s 11

8. Si ngu lar Matrix : A sin g ula

r ma trix is a sq uar e ma trix tha t is
im crs c . A mat rix is sin g ula r o r deg not inv erti ble i.e. it doe s not hav e
an
ene rate if and onl y if its det erm ina
nt is zer o i.e [ ~
9. :\on -~i ngu lar Matrix or Inv
ert ible Ma trix : A squ are ma trix
is non -sin gul ar or ·inv erti ble if its
detcm1 ina nt is no n-zero i.e. JAJ-:t:
0 . A non -sin gul ar ma trix has a ma
tri~ ~n~ t:_f_S~.
IO. Or tho gon al Matrix : A squ are
ma trix is sai d to be ort hog ona l
if A• A.,. = I. ln oth er wo rds the

ll 2 -21
tran .. r o~c o f ort hog onal ma trix is
equ al to the inv ers e of the ma trix
i.e. 1 1
A = A- •

r.x amp Ic : If. /1 - -I - 2I

L. 2I - 21
2 the n A '/ =-
I
2
J I 2
- 2 2 - 1 3
],l 2 - 2 - 1 3 ~]

ilnd
0
01
0 A =A.,.
1
==-1 2 l' 2
I
-21
If ma trix A is ort hog ona l then
l - \, I 3 2 -2 ~IL
(iJ Its inv ers e and tran spo se arc
a lsu orthogonal.
(ii) Its d~t erm ina nt is unity i.e. \A\
=± 1.
(iii ) I A 11 A 7 I= I
C) Copyrf&ht !fea d Offi ce: Al ll4-1 15, S mrit l
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 ACALH MY ,~' Linear Algebra I 1.7

t k n 1tHi:n1 ':\tntrh : :\ sqm1t't' nu11ri~ is said to be hermitian if \A =A 0

/ Where A'1 is the transpose
,,f ,, ,niit_~;\t . _' ~, f m;\t l'i , A. L~·. ~.,/)1
.
'\ )
~I
2I 1/j 3 + 2i
2-3/J
,, l 1i ..."! i then l'<mjugutc or A+~2i 2 - [

·'~ ' /. ~
3 2 +J i 3
J 1 '
~I 1 + Ji
:t , ', l· 2i 2 == A
' .,i -i '.'\

8-Kl'" lkrmithm ,\:\trix : :\ sqtmrc matrix A is said to be skew hermitian if~- ~ =_·A0 J
1 2 4
.·l -l- 2~ 3i ~Jj :/;]
- 4 + ~i 2i - 3i

Hi]
.h.1

( \ '\t\i\l ~;lt;.' l'\(

-i
[
'.
.·l = - -" + .,1
2 + Ji
0 - 2i ⇒ A 0
=
[ -i
2+3i
- 2+3i
0
-4-5i]
-21 =-A
.., .
- 4 - ~i - 2i .)/ .h.1 4-Si -2i 3i 3x3

.-\ ll d1e-~l~gopa l clements of Skew Hem1itian ma~ aree 1U1er zero _Qr pure imaginary
(ii) . .:\ II-the di:rn:o~~~ k mems offkl·mitiar;·~ tri;-are real. ·----.__ \
-----::::.. ----
{iii') l 'pper :lnd lower diagonal elements should be complex conjugate pair.

3.. l'nit:ary ,tatrix : A square matrix is said to be unitary i f ' ~ w h e r e A 9 is transpose of

N r\ju~n-e of matrix A.

£.xample : A =
1+ ;
2
1-i
-l + i
--
2
-1-i
⇒ Ae =
[ 1-i
2
-12-i
I +i
2
-I +i
l ⇒ A·A ' =[1 O
~L
2 2 J x:! 2 2x2

@ Key Point
If matrix A is unitary then
(i) Its inYerse and transpose are also unitary.
(ii) Its detem1inant is unity i.e{I2G_ :ilJ
(iii) ' .-l -4 6 I= l
l4. Periodic ~1atrix : A square matrix is said to be a periodic if
· K is known as perio.d _Qf the matrix.
---=
AK +i A where, K: Positive integer.
__________,,,
is; lnYoluto~· MatrL~ : A matrix is said to be involutory if ~ ? ~

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 1 .8
I . .
Eng1neenng Mathematics
. GATE ACADta...
. . ...,, 1
which, when mult1phed itse f1.e. I _y
b:-;--:.:... •
. • ,

. •0 .

16. Idempo tent :'\'latrn:: An idempotent matnx 1s a squatc; matnx

• •
. • •

=~ · - . . . . . ·I ti1I;' positive 111tee e

0
~ i•e•
A = .-i . A penod1c matnx 1s said to be idempotent " 1en ~

_.(·-1 = A ,41-1= A ⇒⇒
,f =A
•
··1 potent· 1,11. 1atnx .. . ·1 , t 1118...11·ix if there exists a positive 1'nte&er/(
17. ,-~• : A square mamx 1s called a 111 po1en ' • ·
suchtnat .-f;.· = O.
The least positive value of K is called t11e index of nilpotent matrix A.

\ i) DetenninanL9f lde!}1potent marrix is either ,0 or ~

(ii) Determinant and Trace of nilpotent matrix is zero.
(iii) Inverse of nilpotent matrix does not exist.
18. Invertib le Matrix : A matrix A is said to be invertible or non singular or non degenera
te if there
- --- -
exists a matrix B such that AB = BA = I . II

(i) If matrix A is invertible then the inverse is unique.

(ii) If matrix A is invertible then A cannot have a row or column consisting of only zeros.
•'.;:__ 19. Rotation Matrix : A rotation matrix inn-dimensions is an x n special orthogon
al matrix, that is an
orthogonal matrix whose determinant is 1. i.e. ~~ = R- , = I
1
IRI
cos 8 - sin 8]
Example: A = [ .
sm8 cos8
20.Norm al matrix : A matrix is normal if it commutes with its conjugate transpose. A complex
square
matrix A is normal if ;f A = AAr or A A = AA6':
- ----
8

A real square matrix A is normal if ,1 r A = AA r since a real matrix satisfies A= A.

Eigen Value s and Eigen Vecto rs
Chara cterist ic Roots or Eigen Values or Latent roots
Eigen value and Eigen vectors are only valid for square matrix, the roots of characteristic equation
IA -UI = 0 = 0 are called characteristic roots or Eigen values of matrix A.
2 4
Exampl e : A= [ ]
5 3 2x2

The characteristic equation of matrix is given by, IA - Ail =o

2-A 4
=0
5 3- A
(2 - 11.)(3 - A) - 20 = 0 ⇒ '). . - 5')... + 6- 20 = O
2

SA - 14 = 0 ⇒ A
2 2
11. - TA.+ 2A- J4 _ 0 1
-
- ⇒ 11, =7,-2

Head Office: A/114- 115, Smriti Nagar, Bhilai (C.G.) ContIC1 •, 9713113156 9S8989 de .
- - ! ·
Cl Copyright - -- -
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870
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GATE ACADEMY OJ) Linear Algebra I 1.9
- Properties of Eigen Values or Characteristics Roots
(i) The sum or Eigcn values of a matrix is equal to the trace of the mati:_ix w_h~re the sum of the
clements of principal diagonal of a matrix i~ c~lle._~--~~ i,"rac_e.~_f.mat.rix.
L 0\,,) = "-1 + "-2+ "- 3= Trace of matrix
I

(ii) The product of Eigenvalues of a m~~rix A is equal to the determinant of matrix A.

IT
(A,) = "-1"-2"-1= IAI - ·-

(iii) For Hermitian matrix every Eigenvalue is real.

(iv) Every Eigen value of a Unitary matrix has absolute value i.e. i"AI = l
(v) Any square matrix A and its transpose Ar have same Eigenvalues.
(vi) If Ai,A 2 .... .A,, are Eigenvalues of A then Eigenvalues of
✓ KA ar·e K1/\. 1, K1/\. 2 ... ...K"I/\. 11 •
✓ Am are '\/\. m, /\.
'\ m.....,./\. Ill
•
1 2 11

✓ -I 1 I 1 1
A are- ,- , - ..... - ✓ A+KJare "J.. 1 +K,"J.. 2 +K, "A 3 +K ............."A,, +K
A1 "A2 A3 \,

(viii) If A is an Eigen value of an Orthogonal matrix A then ~ is also an Eigen value of

."A

(ix) The Eigenvalue of a symmetric matrix are purely real.

(x) The Eigenvalue of skew-symmetric matrix are either purely imaginary or zero.
(xi) Z~ is an Eigen value of a matrix if and only if a matrix is singular.
(xii) If all Eigenvalues are distinct then the corresponding Eigenvectors are independent.
(xiii)The set of Eigenvalues are called the spectrum of A and the largest Eigenvalue in magnitude
is called the spectral radius of A. Where A is the given matrix.
Cayley-Hamilton Theorem: [ v<f'Y s c,u a,r rn~ l, it- _c, ;:> h,J if~ :i s OLu/)
According to Cayley-Hamilton Theorem,
r h v i JOCr ,~ l·,( t <f 1)<)/.; o n
"Every square matrix satisfies its own characteri_5-ti_c_~quation."
This-theorem is only applicable fo~-square_; atrix. This theorem is used to find the inverse of the
matrix fn tne form of matrix polynomial.
If A be nxn matrix and its characteristic equation is,
a0 ')..," +a 1')..," - 1 + .... +a,, =0
Then, according to Cayley-Hamilton Theorem,
a0 A" +a 1A"-1 + .... +aJ =0 11

Eigen Vectors
If a matrix A having characteristic root A then we have a non-zero vector X which satisfies the
equation [ A _ ')..,J] [ X] =[O] . Where the non-zero vector X is cal led characteristic vector or Eigen
vector.
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 1.1 O Engineering Mathematics GATE ACADEMY®
If there exist Eigenvector X
corresponding to Eig en va lue
"- then the rel~tion for ma trix
by , A is given

AX='"A.X
Properties of Eigen Vecto
rs
(i) For every Eigenvalue the
re exist atleast on e Eig en ve
(ii) If "J,... is an Eig en va lue cto r.
of a matrix A, then the corres
ponding Ei ge nv ec tor Xi s
we have infinite number of no t un iqu e. i.e.
Eigen vectors corresponding
to a single Eigen value.
(iii)lf Ap "J,... , .. .. . A" be distin
ct Eigen values of a n x n ma
2 trix, then corresponding Eig
X P X , ..... X ,, form a linearly en ve cto rs ==
2 independent set.
(iv) If two or more Eig en va
lue s are equal then Eig en ve
cto rs are linearly dependen
(v) Two Eigenvectors X and t.
X are called orthogonal vecto
1 2 rs if ·I)(/ X 2 = 0 .
(vi) A matrix is said to be
defective if it fails to have
n linearly ind ep en de
therefore it is not diagonaizab nt Ei ge n vectors and
le. All defective matrices ha
but not all matrices having ve few er tha n n dis tin ct Eig en values,
fewer than n distinct Eigen
values are de fec tiv e.
Normalized Eigen Vectors
:
A normalized Eigen vector
is an Eigen vector of length
one. Co nsi de r an Eig en ve
cto r X =\al
then length of this Eigen ve
ctor is II X 11= ,Ja2+ b2
lbl xl

a
Normalized Eigen vector is ✓ a2 + b2
X = _! !_ = Ei ge n ve cto r
11 X II Le ng th of Ei ge nv ec tor =
b
,Ja2+ b1
2xl
Ex am ple : X ~ [~ ] , II X II= -,h2 + 7 2 = ✓53

Length of the no rm ali zed Ei

ge n ve cto r is alw ay s unity
.
Echelon fo rm of a M at ri
x:
A matrix A is sai d to be in
Ec he lon for m, if the fol low
ing ho ld:
(i) Ev ery row of ma tri x A,
wh ich ha s all its en tri es zer
entry. o oc cu rs be low ev ery row
wh ich ha s a non-zero \
(ii) Th e fir st no n-z ero en try
in ea ch no n-z ero row is eq
ua l to 1.
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G.), Co nta ct: 9713113156,
Bran ch Office : Raipur, 0
: 79743-90037, Bhopal, 0 : 9589894176 www.gatea~d.= ~ ~
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GATE ACADEMY
®

(iii}1·1ic number of' zer~s preceding the first non-zero elemen t in a row is less
Unear Algebra I 1.11
than the numbe r of such
1/.cro in the succeeding row.
I 5 3 2
0 0 4 6
;1 _ is an Echelo n form.
0 0 0 5
0 0 0 0

Rank of Matr ix :
The rank or a matrix is a numbe r equal to the order of the highest order non-va
nishing minor, that can
be formed from the matrix.
The rank of a matrix is said to be r if,
I. There is at least one non-ze ro minor of order r.
2. Every minor of A having order higher than r is zero.

The rank of a matrix A is the maxim um number of linearly independent column

s or Rows.
A matrix is full rank, if all the rows and columns are linearly independent.
i.e. having rank as
large as possible otherwise, the matrix is rank deficient
Rank of the matrix A is denoted by p(A).

Properties of Rank of Matrix

Rank of the matrix does not change by elementary transformation, we can
calculate the rank by
elementary transfo rmatio ns by changi ng the matrix into echelon form. In echelo
n form, rank of
matrix is numbe r of non-ze ro row of matrix.
The rank of matrix is zero, only when the matrix is a null matrix.
p(A) s min (Row, Colum n)
p(AB) s min [p(A), p(B)]
p(A rA) = p(AAT) = p(A) = p(Ar)
If A and Bare matric es of same order, then p(A + B) s p(A) + p(B) and p(A-B
) ~ p(A)-p (B)
(vii) If A0 is the conjug ate transpo se of A, then p(A 0 ) = p(A) and p(AA
0
) = p(A)
(viii) The rank of a skew symme tric matrix cannot be one.
(ix) If A and Bare two n-rowe d square matrices, then p(AB) ~ p(A) + p(B)-
n•
Solu tion of Line ar Simu ltane ous Equa tions
There are two types of linear simulta neous equations
(i) Linear homog eneous equatio n : AX= 0
(i) Linear non-ho mogen eous equation : AX= B

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 1. 12 Engineering Mathematics
GATE ACADEMY e
Steps to investigate the consistency of system
of linear equations.
1. First represent the equations in matrix
fonn as AX= B.
2. System equation AX= Bis checked for con
sistency as to make Augmented Matrix (A
: B].
Augmented Matrix
(A : B)

Inconsistent Consistent
p(A) -:t p(A: B) p(A) = p(A : B)
Result No solution When p(A) = p(A: B)
= No. of unknown variables
Result : Unique solution
When p(A) = p(A: B)
< No. of unknown variables
Result : Infinite solution
Let A be a rn x n matrix. A homogeneous syst
em of equations AX= 0 will have a unique
the trivial solution X = 0, if and only if rank solution,
(A) = n. In all other cases, it will have infi
many solutions. As a consequence, if n > nitely
rn i.e,, if the number of unknowns
number of equations, then the system will is larger than the
have infinitely many solutions.
W Key Point
The number of linear independe~t infinite
soluti_ons of hom~genous linear equations
n variables is (n- r), where r ts rank of AX = having
matnx A. n - r 1s also the number of pa 0
infinite solution. t . h
rame ers mt e
If A is a square matrix of order n and
(i) \A\=0 then the rows and columns are linearly dependent and system has
a . . .
many solution. non-tnv1al solution or
(ii)\A\*o then the rows and columns are
linearly independent and system h ..
. ue solution
umq . .
as tnvial solution or

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