12

T am il N adu

M AT H EM AT ICS ( AS PE R TH E LATE ST SY LLAB U S)

V O L U M E - 1 M e e nak s h i C h andr as e k ar M .S c .,M

.E d.,M

.P hi l .

Reviewed by

K A nanda K u m ar M .S c .,M

S h ob

M .S c .,M

.E d.,M &

.P hi l .

ana V e l r aj .E d.,M

.P hi l .

FULL MARKS PVT LTD (Progressive Educational Publishers)

CHENNAI - 600 017

Published by:

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R E V IS E D E D IT IO N

“ T ih s bok i s m e ant f or e duc at i ona l and l e ar ni ng pur pos al l r e as ona bl e c ar e t o e ns ur e t ha t t he c ont e nt s of t he bok ot he r i nt e l l e c t u al pr o pe r t y r i ght s of any p e r s on i n any m ha s / ha ve be e n una bl e t o t r ac k any s our c e and i f any pl e as e not i f y t he publ i s he r i n w r i t i ng f or c or r e c t i ve ac

e s . T eh aut hor ( s ) of t he bok ha s / ha ve do not vi lo at e any e xi s t i ng c opyr anne r w ha t s oe ev r . I n t he e ev nt t he aut c opyr i ght ha s be e n i nadve r t e nt l y i nf t i on.”

t ake i ght hor r i nge

or

n

(s) d,

N ote f r om D e ar m

e m b e r s of

t h e l e ar ni ngf

th e P ub lish er s

r at e r ni t y ,

W ar m gr e e t i ng s f r om t he P ubl i s he r s of F u l l M ar k s . T he M at h e m at i c s X I I gui de i s r e pl e t e w i t h s uc c e s s s t r at e gi e s and t i ps f or de ev l opm e nt of s c hol ar l y at t i t ude and appl i c at i on s ki l l s f or s t ude nt s . I t ha s be e n w r i t t e n ac c ro di ng t o t h e l at e s t s yl l abus and t e xt bok i s s ue d yb t eh T am i l N adu S t at e B oa r d. T ih s s upor t b ok pr ovi de s c om pl e t e t ut or i al s upor t t o s t ude nt s t o c ove r t he c our s e c ont e nt s easily, develop confidence and answer the various types of uestions generally as ed by the B oa r d. S al i e nt F e at u r e s of The T op

i c s C ove

t h e b ok r e d ha ve

: be e n m e nt i one d at t he

be gi nni ng of

e ve r y c ha pt e r .

ach chapter contains the C h ap t e r M ap , w hi c h gi ve s an i de a about r e ade r f oc us on t he i m por t ant he adi ngs .

t he

c ont e nt s and he l ps

omplete summary of the chapter is provided in the I m p or t ant P oi nt s t oR ll the T e xt u al Q u e s t i on s h ave eb e n s ol ve d and a l ar ge have been provided wherever re uired. I m p or t ant F or m u l ae ha ve

be e n i nc l ude d f or

For M u l t i p l e C h io c e Q u e s t i on s ( w eh r e ve r appl i c abl e .

num

be r of

t he

e m e m b e r .

A ddi t i on al Q u e s t i o ns

a c l e ar unde r s t andi ng.

s), S ol u t i on s or

H i nt s ha ve

al s o be e n pr ovi

de d

separate section on the Common Errors and their Rectifications hi ghl i ght s t he ar e as w eh r e s t ude nt s ge ne r al l y ge t c onf us e d and m ake m i s t ake s w hi l e unde r s t andi ng t he c onc e pt s or e xe r c i s e s . The boo is a up to date, dependable and learner friendly resource. W e r e gi s t e r our s i nc e r e gr at i t u de t o t he aut orh and t he the boo in record time with remar able uality.

s c hol

ar l y e di t or s f or

he l pi ng us

br i ng out

onstructive suggestions for improvement of the uality of this boo are most welcome.

( iii)

T op

4

anc e .

i c s C ove

r e d

I m p or t ant D i agr am s  z +1 R P of   =  z+i  i . e . , x (x + 1) + y (y + 1) = x2 + x + y2 +y = ⇒ x + y – 2y – 1 =

I nv er se T r igonom etr ic F u nctions

P olar and E u ler f or m of a C om plex N u m b er : P olar f or m of a com plex nu m b er :

TOPICS COVERED 1 . I ntrod u ction 3 . S ine F u nction and I nv erse S ine F u nction 5 . T h e T angent F u nction and th e I nv erse T angent F u nction 7 . T h e S ecant F u nction and I nv erse S ecant F u nction 9 . P rincipal Valu e of I nv erse T rigonom etric 1 F u nctions

x( x + 1) + y ( y + 1) ⇒ =1 x 2 + ( y + 1) 2 x2 + (y + 1)2 x2 + y2 + 2y +1 0; i . e ., x – y –1 = 0

1

2 . S om e F u nd am ental C oncepts 4 . T h e cosine fu nction and I nv erse cosine fu nction 6 . T h e C osecant F u nction and th e I nv erse C osecant F u nction 8 . T h e C otangent F u nction and th e I nv erse C otangent F u nction 0 . P roperties of I nv erse T rigonom etric F u nctions

y

P (x, y)

P (x, y)

P (r , θ ) 2

iy x+

2

θ x O

CHAPTER MAP

+

y

y = r sin θ

θ x = r cos θ O

O

R ectangular coordinates

x

=

r

r

T h e c h ap t e r at agl

M

S uperimpose polar coordinates on rectangular coordinates

P olar coordinates

x = r cos ...(1) y = r sin ...(2) A ny non- z ero complex number z = x + i y can be ex pressed as z = r cos + i r sin D ef inition: L et r and e polar coordinates of the point (x, y) that corresponds to a non- z ero complex number z = x + i y. The polar form or trigonometric form of a complex number P is z = r (cos + i sin ). P r incipal A r gu m ent of a com plex nu m b er I I - Q uadrant

I - Q uadrant y

y

θ=α

I V - Q uadrant

I I I - Q uadrant y

θ=p α

y

p

θ =α

θ = –α

z z θ

α

α x O

O

x

α

O

x

θ

O

θ=α

θ=p α

x

α z

z θ =α

p

θ = –α

S ome of the properties of arguments are (i ) arg (z1 z2) = arg z1 + arg z2

I m p or t ant E q u at i on s (i) Centre (h, k)

C h ap t e r M

Equation of a circle in general form is x2 + y2 + 2gx + 2 fy + c = 0 .

1

(i) centre (–g,– f ) (ii) radius =

g

2

ap

(ii) radius ‘ r ’

+ f

2

−c

The circle through the intersection of the line lx + my + n = 0 and the circle

A pplications of M atr ices and D eter m inants

x2 + y2 + 2gx + 2 fy + c = 0 is x2 + y2 + 2gx + 2 fy + c + λ (lx + my + n) = 0, λ ∈ R'

TOPICS COVERED

Equation of a circle with (x1 , y1 ) and (x2 , y2 ) as extremities of one of the diameters is

1 . I ntrod u ction 2 . I nv erse of a N on- S ingu lar S q u are M atrix

(x − x1 )(x − x2 ) + ( y − y1 )( y − y2 ) = 0 .

3 . E lem entary T ransform ations of a M atrix 4 . Applications of M atrices: S olv ing S y stem

Equation of tangent at (x1 , y1 ) on circle x2 + y2 + 2gx + 2fy + c = 0 is

xx1 + yy1 + g ( x + x1 ) + f ( y + y1 ) + c = 0 Equation of normal at (x1 , y1 ) on circle

x2

+

y2

of L inear E q u ations

5 . Applications of M atrices: C onsistency of sy stem

of linear eq u ations by rank m eth od

CHAPTER MAP

+ 2gx + 2fy + c = 0 is

A pplications of M atr ices and D eter m inants

yx1 – xy1 + g ( y – y1 ) – f ( x – x1) = 0

Tangent and normal Curve

Equation

Equation of tangent

Circle

x2 + y2 = a2

(i) cartesian form xx1 + yy1 = a2 (ii) parametric form x cos θ + y sin θ = a

(i) cartesian form xy1 – yx1 = 0 (ii) parametric form x sin θ – y cos θ = 0

Parabola

y2 = 4ax

(i) yy1 = 2a ( x + x1) (ii) yt = x + at2

(i) xy1 + 2y = 2ay1 + x1y1 (ii) y + xt = at3 + 2at

xx1

Equation of normal

yy1

Ellipse

x2 y2 + =1 a 2 b2

a 2 x b2 y (i) 2 + 2 = 1 + = a 2 − b2 (i) a b x1 y1 (ii) x c o s θ + y s i n θ = 1 (ii) ax − by = a 2 − b 2 c o s θ s i nθ a b

Hyperbola

x2 y 2 − =1 a 2 b2

yy a 2 x b2 y − 21 = 1 (i) + = a 2 + b2 a2 b x1 y1 x s e c θ y t an θ ax by (ii) − = 1 (ii) + = a 2 + b2 a b s e c θ t an θ (i)

xx1

I nverse of a N on- S ingular S quare M atrix

E lementary transformations of a M atrix

S olving system of L inear E quations

Consistency of system of linear equations by rank method

A dj oint of a square matrix

R ow - E chelon form

S ystem of L inear equations in M atrix form

N on homogeneous linear equations

P roperties of I nverses of matrices

R ank of a matrix

M atrix I nversion M ethod

H omogeneous S ystem of linear equations

A pplication of matrices to geometry

G auss - J ordan M ethod

Cramer' s R ule

A pplication of matrices to Cryptography

( iv )

G aussian E limination M ethod

I m p or t ant P oi nt s t oR

e m e m b e r

T e xt u al E xe r c i s e s E X E R C IS E - 1 .3

I m por tant P oints to R em em b er

. ol e e ollo n s s e ( i) 2x + 5 y = – 2, x + 2y ( iii) 2x + 3 y – z = 9 , x + y + ( iv) x + y + z – 2 = 0 , 6 x – 4 Sol. (i ) 2x + y = −2, x + 2y =

a ons a r n ers on e od ii) 2x – y = 8 , 3 x + 2y = – 2 z= 9 , 3 x– y– z= – 1 y + 5 z – 3 1 = 0 , 5 x + 2y + 2z = 1 3 3  2 5  x   −2  The matrix form of the above equations is    =    1 2  y   −3 

v A dj

oint of a square matrix A =Transpose of the cofactor matrix of A v A (adj A ) = (adj A )A =| A | I n −1

v A

=

1 A

adj A .

1 1 −1 −1 T = . (i i ) A T = A −1 (i i i ) ( λA ) = A λ A B )−1 = B −1A −1 (i i ) (A −1)−1 = A v I f A is a non- singular square matrix of order n , then 1 −1 −1 v (i ) ( adj A ) = adj A = A (i i ) adj A = A n A A

v (i )

( ) ( )

−1

−1

where λ is a non- z ero scalar.

(i . e )

v (i ) (A

( )

n −2

(i i i ) adj (adj A ) = A (v i i ) adj ( A B v (i

 2 5 A =  ; 1 2

N ow

−1

adj (A ), λ is a nonz ero scalar

o find A–

( )

A X =B

2 A =  1 2 |A | = 1 1

(v i ) ( adj A ) = adj A T T

) = (adj B )(adj A )

=±

) A

(i v ) adj (λA )= λ n A

( n −1)2

(v ) adj (adj A ) = A

A X =B

W here −1

(i i ) A = ±

adj A .

adj A

1

adj ( adj A

adj A

).

) A matrix A is orthogonal if A A T = A T A = I (i i ) A matrix A is orthogonal if and only if A is non- singular and A −1 = A T v M ethods to solve the system of linear equations A X = B (i ) B y matrix inversion method X = A −1 , A 0 ∆ ∆ ∆ (i i ) B y Cramer’ s rule x = 1 , y = 2 , z = 3 , ∆ ≠ 0. ∆ ∆ ∆ (i i i ) B y G aussian elimination method v (i ) I f (A ) = ([A | B ]) = number of unk nowns, then the system has unique solution. (i i ) I f (A ) = ( A ) n m er of nkno ns, then the system has infinitely many solutions. (i i i ) I f (A ) ([A | B ]) then the system is inconsistent and has no solution. v The homogeno s system of linear e ations A = 0 (i ) has the trivial solution, if | A | 0. (i i ) has a non tri ial sol tion, if A = 0.

 x 2 1   =A B = 1  y ⇒ x = 11; y = 4 (i i ) 2x y = ; 3x + 2y = 2

N ow

o find A–

Sol. L et N ow ∴ O perating R O perating R

1

2

O perating R

2

O perating R

1

1

2

1 R 3

2

3

1

2

Sol. L et

N ow O perating R

2

∴

2

2 and

R

3

1  1 0  1 −1 0 3 −5 =  −2 1    0 2 0   −1 0 1 R 2 O perating R 2 3 1  1 −1  1 0 1 −5 / 3    =  −2 / 3 0  0 2  −1

3

e n erse o

=

|A |

|A | =

a=

1B 2 −1 =4+3= 3 2

1  2 (adj A ) =

1  2 1   7  −3 2 

Rectifications

8 =2 4

a t1 t2 = 2(t)(3t) = 6t2 a(t1 + t2) = 2(t + 3t) = 8t Point = (6t2, 8t)2 Common Errors and its Rectifications

Common Errors Rectifications 1. In general form of a circle tak ing centre as 1. Centre is (–g, –f ) (g, f ) wrongly 2. Tak ing radius as

g

2

+ f

2

+c

3. In conics tak ing e value wrongly

wrongly

2. Radius =

g

2

+ f

3. For an Ellipse e =

2

−c a

2

and for Hyperbola e =

 2 −7 / 3 1  6 −7  1=  −1 4 / 3  =   3  −3 4 

A =IA 2

So,

1 0   2 −7 / 3   =  −1 4 / 3  A   0 1 

rans or a ons find 1 −1 1    A =  2 1 −3 1 1 1 

1

Here, t1 = t, t2 = 3t,

4 7  . 3 6

 1 −1 1 1   =  −3 4  A    0 3

A

s n ele en ar

ar

−1  1 1 1 0 1 =  −1 4 / 3 A    

2

1

n

1 −1 1 1  3 6  =  0 1  A    

Thus .

A

1

−1  2

Common Errors

A d d itio n a l Q u e s tio n s S o lv e d e ollo

 −2   4 − 15   −11   =  =    4   −3   −2 + 6 

⇒ X =A

2 A =  3 2 adj A =   −3

n

u e s t i on s

rans or a ons find e n erse o 4 7 A =   3 6 A = IA 4 7 1 0   3 6  = 0 1  A    

0

5 2

A X =B

1

∴

s n ele en ar

= 1

 2 −1  x   8  Sol. The above equations in matrix form is    =    3 2   y   −2  (i . e ) A X =B 8  x  2 −1 W here A =   ; =  y  and B =  −2      3 2 

eore

.

5  2 5 =4 2

∴

A d j oint of a Sq u ar e M atr ix efin on L et A be a square matrix of order n . Then the matri of cofactors of A is defined as the matrix obtained by replacing each element a i j of A with the corresponding cofactor A i j . The adj oint matri of A is defined as the transpose of the matri of cofactors of A. t is denoted y ad A.

A ddi t i o nal Q

 −2   x =   and B =    −3   y ⇒ X = A 1B

 2 −5  adj A =    −1 2  1 1  2 −5   −2 5  (adj A ) = A 1=    = |A | −1  −1 2   1 −2 

v (i

For every square matrix A of order n , A (adj A ) = (adj A )A = A I

o l near e

e

ar

1 −1 1    ∴  2 1 −3 = 1 1 1 

− b2 a

2

a

2

+ b2 a

2

4. Tak ing the condition for y = mx + c to be a 4. The condition for y = mx + c to be a tangent to tangent to the conics wrongly 1) Ellipse is c2 = a2 m2 + b2 2) Hyperbola is c2 = a2 m2 – b2 5. Tak ing the point of contact wrongly 5. The point of contact  −a 2 m b2  ,  to ellipse is  c   c

1 −1 1   2 1 −3  . 1 1 1 

1 0 0  0 1 0    A 0 0 1 

 − a 2 m −b 2  , and hyperbola is   c   c

1

0 0 A  1

0 0 1 / 3 0  A 0 1 

(v )

Contents CH AP T ER-1 A p p l i c at i on s of

M

at r i c e s an dD e t e r m i nant s .

7

CH AP T ER-2 C om

p l e xN

u m b e r s .

10 .

CH AP T ER-3 T h e or yof

E q u at i on s .

160 .

CH AP T ER-4 I nve r s e T r i gon

om

e t r i c F u nc t i on s .

197 .

CH AP T ER-5 T w oD

i m e ns i on al A nal y t i c al G

e om

e tr y -II .

CH AP T ER-6

234

A p p l i c at i on s of

V e c t or

A l ge b r a .

301

( v i)

1

A pplications of M atr ices and D eter m inants

TOPICS COVERED 1 . I ntrod u ction 2 . I nve rse of a N on- S ingu lar S q u are M atrix 3 . E lem entary T ransform ations of a M atrix 4 . Applications of M atrices: S olvi ng S yst em

of L inear E q u ations

5 . Applications of M atrices: C onsistency of sys tem

of linear eq u ations by rank m eth od

CHAPTER MAP A pplications of M atr ices and D eter m inants

I nverse of a N on- S ingular S quare M atrix

E lementary transformations of a M atrix

S olving system of L inear E quations

Consistency of system of linear equations by rank m ethod

A dj oint of a square matrix

R ow - E chelon form

S ystem of L inear equations in M atrix f orm

N on homogeneous linear equations

P roperties of I nverses of matrices

R ank of a matrix

M atrix I nversion M ethod

H omogeneous S ystem of linear equations

A pplication of matrices to geometry

G auss - J ordan M ethod

Cramer' s R ule

A pplication of matrices to Cryptography

7

G aussian E limination M ethod

I m por tant P oints to R em em b er v A

dj oint of a square matrix A =Transpose of the cofactor matrix of A v A ( adj A ) = ( adj A ) A =| A | I n v A

−1

=

1 A

adj A .

1 1 −1 −1 −1 T (i i i ) ( λA ) = . (i i ) A T A = A λ A v (i ) (A B )−1 = B −1A −1 (i i ) (A −1)−1 = A v I f A is a non- singular square matrix of order n , then 1 −1 −1 v (i ) ( adj A ) = adj A = A (i i ) adj A = A A v (i )

( ) ( )

A −1 =

−1

( )

(i i i ) adj ( adj A ) = A (v ) adj ( adj A ) = A (v i i ) adj ( A B v (i

) A

−1

n −2

A

( n −1) 2

, where λ is a non- z ero scalar.

n −1

(i v ) adj ( λA ) = λ n

−1

adj ( A ) , λ i s a no nz e r o s c al ar

( )

(v i ) ( adj A ) = adj A T T

) = (adj B )(adj A )

=±

1 adj A

adj A .

(i i ) A = ±

1

adj

adj A

(adj A ) .

) A matrix A is orthogonal if A A T = A T A = I (i i ) A matrix A is orthogonal if and only if A is non- singular and A −1 = A T v M ethods to solve the system of linear equations A X = B (i ) B y matrix i nversion method X = A −1 , A 0 ∆ ∆ ∆ (i i ) B y Cramer’ s rule x = 1 , y = 2 , z = 3 , ∆ ≠ 0. ∆ ∆ ∆ (i i i ) B y G aussian elimination method v (i ) I f (A ) = ([A | B ]) = number of unknow ns, then the system has unique solution. (i i ) I f (A ) = ( A ) n m er of nkno ns, then the system has infinitely many solutions. (i i i ) I f (A ) ([A | B ]) then the system is inconsistent and has no solution. v The homogeno s system of linear e ations A = 0 (i ) has the trivial solution, if | A | 0. (i i ) has a non tri ial sol tion, if A = 0. v (i

A d j oint of a Sq u ar e M atr ix efin on L et A be a square matrix of order n . Then the matri of cofactors of A is defined as the matrix obtained by replacing each element a i j of A with the corresponding cofactor A i j . The adj oint matri of A is defined as the transpose of the matri of cofactors of A. t is denoted y ad A. eore

8

For every square matrix A of order n , A ( adj A ) = ( adj A ) A = A M ath em atics – 1 2

I n

P r oper ties of inve r ses of m atr ices I f A is non- singular, then 1 −1 −1 = (i ) A (i i ) A T = A A

( ) ( )

−1 T

(i i i ) ( λA

)−1 =

1 A λ

−1

where λ is a non- z ero scalar.

L ef t C ancellation L aw L et A , B , and C be square matrices of order n . I f A is non- singular and A B = A C, then B = C. R ight C ancellation L aw L et A , B , and C be square matrices of order n . I f A is non- singular and B A = CA , then B = C. R eve r sal L aw f or I nve r ses I f A and B are non- singular matrices of the same order, then the product A B is also non- singular and (A B )−1 = B −1A −1. L aw of D ou b le I nve r se I f A is non- singular, then A −1 is also non- singular and (A −1)−1 = A . eore I f A is a non- singular square matrix of order n , then 1 −1 −1 = A (i i ) adj A (i ) ( adj A ) = adj A A

( )

(i i i ) adj ( adj A ) = A (v ) adj ( adj A ) = A

n −2

= A

n −1

(i v ) adj ( λA ) = λ n A

(v i ) ( adj A

( n −1) 2

)T

−1

adj ( A ) , λ i s a no nz e r o s c al ar

( )

= adj A

T

A pplication of M atr ices to G eom etr y efin on A square matrix A is called orthogonal if A A

T

=A

TA

=I .

A pplication of M atr ices to C r yp togr aphy

O ne of the important applications of inverse of a non- singular square matrix is in cryptography. Cryptography is an art of communication between two people by k eeping the information not k nown to others. I t is based upon two factors, namely encryption and decryption. E ncryption means the process of transformation of an information (plain form) into an unreadable form (coded form). O n the other hand, D ecryption means the transformation of the coded message back into original form. E ncryption and decryption require a secret technique which is know n only to the sender and the receiver. This secret is called a ke y. O ne way of generating a ke y is by using a non- singular matrix to encrypt a message by the sender. The receiver decodes (decrypts) the message to retrieve the original message by using the inverse of the matrix. The matrix used for encryption is called encryption matrix (encoding matrix) and that used for decoding is called decryption matrix (decoding matrix) .

Applications of M atrices and D eterm inants

9

E X E R C IS E - 1 .1

.

nd

e ad o n o

 −3 ( i)  6  −3 Sol. (i )  6

4 2  4 2 

e ollo n 2 3 1   ( ii)  3 4 1   3 7 2 

 2 2 1 1 −2 1 2  ( iii)  3  1 −2 2 

 a b I f matrix A =  ,  c d  Then adj A =

(A ) ij

T

 d − b = a  −c

 3 4  2 −4 , S o adj A =  H ere A =   6 2    −6 −3

Sol.

2 3 1   (i i )  3 4 1   3 7 2 

2 3 1   L et A =  3 4 1  adj A = (A  3 7 2 

N ow A

ij

 4 +  7  3 = −  7  + 3  4

1 2

−

3 1 3 2

1 2

+

2 1 3 2

1 1

−

2 1 3 1

 + ( 8 − 7) =  − ( 6 − 7)  + ( 3 − 4)

1 0

− ( 6 − 3) + ( 4 − 3) − ( 2 − 3) T

ij)

T

where (A

= Co- factor matrix of A

3 4  3 7 2 3  − 3 7  2 3 + 3 4  +

+ ( 21 − 12) − ( 14 − 9) + ( 8 − 9)

  1 −3 9   =  1 1 −5      −1 1 −1

∴

 1 −3 9  1 1 −1 T   (A i j ) = 1 1 −5 =  −3 1 1      −1 1 −1  9 −5 −1

(i .e )

 1 1 −1 adj A =  −3 1 1    9 −5 −1

M ath em atics – 1 2

ij)

2 2 1 (i i i )  −2 1 3  1 −2 Sol.

1 2  2 

 2 2 1  2 / 3 2 / 3 1/ 1  −2 1 2 (i .e ) A =  −2 / 3 L et A = 1/ 3 2 /   3  1 −2 2  1 / 3 −2 / 3 2 /  1  + −2  1 2 (A i j ) =  − 9  −2  2  +  1

2 2 1 2 1 2

 + ( 2 + 4) 1 − ( 4 + 2) = 9  + ( 4 − 1)

−

−

−2 1  1 −2   2 2  −  1 −2  2 2  + −2 1 

−2 2 1 2

+

3 3  3

+

2 1 1 2

2 1 −2 2

− ( −4 − 2) + ( 4 − 1) −( 4 + 2)

+ ( 4 − 1) − ( −4 − 2) + ( 2 + 4)

   

 6 6 3  2 2 1 1 1  −6 3 6 = −2 1 2 =  3  9  3 −6 6  1 −2 2 S o adj A = ( A .

nd

e n erse

e ss 5 ( ii) 1 1

 −2 4  ( i)    1 −3  −2 4  Sol. (i )    1 −3 For a matrix A , A and so A

1 does

ij )

−1

=

1 A

T

o 1 5 1

1  2 −2 1 = 2 1 −2  3  1 2 2 e ollo n 2 3 1 1    1 ( iii)  3 4 1   3 7 2  5

( adj A ) . W here | A | ≠ 0. f A = 0 then A is called a sing lar matri

not exi st.  −2 4  L et A =    1 −3 −2 4 |A | = =6–4=2 1 −3

S o A is a non singular matrix a nd so A

1 exi

0

sts

Applications of M atrices and D eterm inants

1 1

A For a matrix

1

( adj A ) |A |  a b d A =  , adj A =    c d  −c 1

=

− b a 

1 1  −3 −4  −3 −4 1= ( adj A ) = S o adj A =  ∴ A |A| 2  −1 −2  −1 −2 5 1 1  (i i ) 1 5 1  1 1 5 5 1 1  Sol. L et A = 1 5 1  1 1 5 5 1 1 N ow | A | = 1 5 1 = 5( 25 − 1) − 1( 5 − 1) + 1( 1 − 5) 1 1 5 = (24) 4 4 = 120 = 112 1 ⇒ A is a non singular matrix a nd so A exi sts. 1 N ow A 1 = ( adj A ) |A | adj A = (A i j )T  5 +  1  1 (A i j ) =  −  1  + 1  5

1 5

−

1 1 1 5

1 5

5 1 + 1 5

1 1

−

5 1 1 1

0

1 5  1 1  + ( 25 − 1) 5 1   = −( 5 − 1) − 1 1    + ( 1 − 5) 5 1 + 1 5  +

−( 5 − 1) + ( 25 − 1) − ( 5 − 1)

 24 −4 −4 =  −4 24 −4    −4 −4 24  24 −4 −4 T adj A = ( A ij ) =  −4 24 −4    −4 −4 24  24 −4 −4  6 −1 −1 1 1  1  1  S o A = −4 24 −4 = −1 6 −1 ( adj A ) =  28   A 112   −4 −4 24  −1 −1 6  1 2

M ath em atics – 1 2

+ ( 1 − 5) −( 5 − 1) + ( 25 − 1)

   

2 3 1   (i i i )  3 4 1   3 7 2  Sol.

S oA

1

2 3 1   L et A =  3 4 1   3 7 2  2 3 1 | A | = 3 4 1 = 2( ) 3( 3) + 1(21 3 7 2 = 2(1) 3(3) + 1( ) = 2 + =2 0

exi sts

1

N ow A

=

1 |A |

( adj A )

 1 1 −1   1 1 adj A =  −3  9 −5 −1  

. Sol.

efer 1. (i i )]

 1 1 −1 1  1= −3 1 1 2    9 −5 −1

S o A  c os α  0   − s i n α L et

12)

0 s i nα 1 0  s o  0 c os α  A

a

– 1

F( )

S o [F( )] 1 = A

1

 c os α 0 s i n α  N ow A =  0 1 0     − s i n α 0 c os α  c os α 0 s i n α 0 1 0 |A | = − s i n α 0 c os α E xpa nding the determinant - along R 0( ) + 1 S oA

1

cos2 α +

sin2

0( ) = 1

exi sts N ow A

1

=

1 |A |

2

W e get

0

( adj A ) =

1 (adj A 1

)

= adj A

Applications of M atrices and D eterm inants

1 3

nd ad A adj A = (A i j )T  1 +  0  0 (A i j ) =  −  0  + 0  1  + ( c os =  − (  + ( − s

o

(i .e ) A

1=

0

−

c os α s i nα c os α

− s i n α c os α

+

s i nα 0

A

Sol.

A

2=

( )

1=

A × A =

L H S =A 1 4

M ath em atics – 1 2

−s i nα c os α −s i nα

c os α s i n α 0 0

+

c os α 0

− ( 0) + ( 1) −( 0 )

+( s i n α) − ( 0) + ( c os α )

 c so α 0 − s i n α   0  1 0   c os α   s i nα 0

5 3 2  −1 −2  show that A – 3A   5 3 A =    −1 −2 

∴

0

−

 c os α 0 s i n α   1 0  i en ( ) =  0   − s i n α 0 c os α  0 s i n( −α )  c os ( −α )  0 1 0 So ( ) =   − s i n( −α ) 0 c os ( −α ) cos α 0 − sin α   1 0  =  0   sin α 0 cos α  (∴ cos ( ) = cos and sin ( ) = sin ) ere (1) = (2) ⇒ ( ) 1 = ( ) .

+

c os α s i n α − s i n α c os α

−

α)

0

  c os α 0 = 0 0) 1     − s i n α 0 i n α)  c os α 0 − s i n α  1) = (A T   1 0 ij) =  0  c os α   s i nα 0

adj A = (A

∴

0

2

3A

– 7I

2

2.

3 2

...(1)

    ...(2)

ence find A .

 5 3   25 − 3   =  −1 −2   −5 + 2 5 3  22 9  = 3     2  −1 −2   −3 1 

5 1

1  0  0   0   0  1  s i nα 0   c os α 

15 − 6   22 9   =  −3 + 4   −3 1  1 0    0 1

0  22 9  15 9  −7 =  − +    −3 1  −3 −6  0 −7  22 =   −3 0 =  0

T o F ind A – 1 N ow we have proved that A 2 P ost multiply by A 1 we get A 3 A 1= O 2

3A

2

=O

2

3  −3 0   2 3   5 3 1 0   5 A 1 = A − 3I =  − 3 = + =     −1 −2  0 1  −1 −2  0 −3  −1 −5 

⇒ (i.e) .

9   −15 −9  −7 0   22 − 15 − 7 9 − 9 + 0  + =  +  1   3 6  0 −7  −3 + 3 + 0 1+ 6 − 7  0   =O 2=R H S 0 

 −8 1 4 9  1

A

Sol.

2 3 A 1=    −1 −5  1 4 4 7  pr ove that A −8 4  1  −8 1 A =  4 4 9  1 −8

1

⇒ – 1

=

1 2 3   7  −1 −5 

AT . 4  7 4 

1 4 −8 1 1  −8 (16 + 56) − 1(16 − 7) + 4 ( −32 − 4)  |A | = 3 4 4 7 = 9 729  1 −8 4 1 1 = [ −8 ( 72) − 1( 9) + 4( −36) ] = 729 [9( −8 × 8 − 1 − 4 × 4) ] 729 1 1 ( −64 − 1 − 16) = ( −8 1) = 1 0 = 81 81 ∴A

1

exi sts

o find ad A

1

( adj A ) |A | adj A = (A i j )T A

1

=

 +  1  (A i j ) = 2  − 9   + 

4 −8

7 4

−

4 7 1 4

1 −8

4 4

+

−8 1

4 4

−

1 4 4 7

−

−8 4

4 7

+

+

4 4  1 −8  −8 1  1 −8   −8 1 4 4 

Applications of M atrices and D eterm inants

1 5

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