Matrices inter first year important

7/28/2019 Matrices inter first year important

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Matrices

VSAQ

. Define trace of the matrix and find trace of

Sol: sum of elements in the principle diagonal of a

square matrix is called trace of A.

=1+ (-1) +1=1.

. Define a symmetric matrix, IfA= , then find x.Sol: A square matrix A is said to be symmetric matrix if

A=A.

x=6.

. Define skew symmetric,If

Matrix, then find x.

Sol: A square matrix A is said to be skew symmetric

matrix

if A=-A

x=-4

. If A= Sol: 1 (3x+24) =45 3x=45-24 3x=21 x=7

. If Find the value of x, y, z and a.Sol: Equating the corresponding elements x=5+3|| 2y=2+8 || z=-2-2 ||a=6+4 x=8, y=5, z=-4 and a=10

. If

Sol: A= A+A= = AA=

=

= 7. IfA= Sol: 2X=B-A2X= X=

8.

Find the cofactor of the elements 2, -5 in the

matrix Sol: the cofactor of the elements 2 is = the cofactor of the elements -5 is =

=-(2-5)

=-(-3)

=3.

9. Find the det of Sol:

=-31-4(-44) +9(-17)

=-31+176-153

=176-184

=-8.

10.If A Sol:A =0 -2-k=0k=-2.

11.Find the ad joint and inverse of ,


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Matrices

Sol:

ifA= =

= = 2.If

=0.

3.If A= Sol:A= =

4.Find the rank of the matrix Sol: = =

Rank of A=3(number of non -zero rows)

5.Construct a matrix A= .Let A= .

A=

16.If A= A= . (1)A+B=

(2)

From (1) & (2)

SAQ

1. If A= .Sol: A=

= =

= = 2. If A=

Sol:


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Matrices

=

= = =

or . If A=

Sol: = -4+1+4

=1

A

. Find the inverse of diag [a b c].Sol: let A= =

5. If I= Sol: = = =

= =.

6. If A= Sol: let p (n) = Step-1: put n=1

A= Step-2: let us assume that p(k) is true for n=k

Put n=k Step-3: put n=k+1

= 7. IfA= (h/w)

Sol: s(n) is true for n=k+1.


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Matrices

. Show that Sol: = = = = =

. Show that Sol: (h/w)

0. Show that Sol: L.H.S = = = e

1. Show that

Sol: L.H.S

= = = =

12. If , then show that

Sol: =cos =-sin

L.H.S = = =


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Matrices

LAQ (2).

Sol: L.H.S

= = R.H.S

.

L.H.S

=

=

=

(2)

3. If

Sol:

{ }

{

}


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Matrices

. Sol: =

= Expanding along =

.

Sol:

=

= Expanding along =

6. S.T (a-b) (b-c) (c-a) (ab+bc+ca).Sol: L.H.S = =

= (a-b) (b-c)

= (a-b) (b-c)

= b(c-a) +(c-a (c-a = (a-b) (b-c) = (a-b) (b-c)

Expanding along = (a-b) (b-c) = (a-b) (b-c)

= (a-b) (b-c) (c-a) (ab+bc+ca).


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Matrices

7. S.T (a-b) (b-c) (c-a)Sol:

= = =

= Expanding along == (a-b) (b-c) (c-a)

. .

: . .

x=4.. If A=

Sol: A=

Now A. adjA= =

=

= = detA.IA. =ISimilarly we can prove that

= I


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Matrices

0. Solve the following equations by using Cramers ruleand matrix inversion method.

a) b)

(a).

Sol: let A= , X= and B=

=3 -4 +5 =3(-7+16) -4(14-40) +5(-4+5)

=3(9)-4(-26) +5(1)

=27+10+5

=136 =18

-4

+5

=18(-7+16) -4(91-160) +5(-26+20)=18(9)-4(-69) +5(-6)

=162+276-30

=408

=3 -18 +5 =3(91-160) -18(14-40) +5(40-65)

=3(-69)-18(-26) +5(-25)

=-207+468 -125

=136

=3 -4 +18 =3(-20+26) -4(40-65) +5(-4+5)

=3(6)-4(-25)+5(1)

=18+100+18

=136

x=

, y=

=

, z=

x=3, y=1 and z=1.(b).

Sol: let A= , X= and B=

=2 =2(1+1) +1(1-1) +3(-1-1)=2(2) +1(0) +3(-2)

=4-6 = -2 =6 =9(1+1) +1(6-2) +3(-6-2)

=9(2) +1(4) +3(-8)

=18+4-24=22-24=-2

=2 =2(6-2) -9(1-1) +3(2-6)=2(4) +1(0) +3(-24)

=8-12=-4

=2 =2(2+6) +1(2-6) +9(-1-1)

=2(8) +1(-4) +9(-2)

=16-4-18

=16-22=-6

x= =1, y= = , z= x=1, y=2 and z=3.(c). A= , X=

and B=

=1 =1(-5-7) -1(-2-14) +1(2-10)

=1(-12) -1(-16) +1(-8)

=-12+16-8=- 4 =9 =9(-5-7) -1(-52-0) +1(52-0)=9(-12) -1(-52) +1(52)

=-108+52+52

=-108+104=-4

= =1 =1(-52-0) -9(-2-14) +1(0-104)

=1(-52) -9(-16) +1(-104)


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Matrices

=-52+144-104

=-156+144=-12

=1 =1(0-52) -1(0-104) +9(2-10)

=1(-52) -1(-104) +9(-8)=-52+104-72

=-124+104=-20x= =1, y= = , z= x=1, y=3 and z=5.(d). A= , X=

and B=

=2 =2(-8-1) +1(4-3) +3(-1-6)

=2(-9) +1(1) +3(-7)

=--18+1-21

=-38

=8 =8(-8-1) +1(-16-0) +3(4-0)

=8(-9) +1(-16) +3(4)

=-72-16+12

=-76

=2 =2(-16-0) -8(4-3) +3(0-12)

=2(-16) -8(1) +3(-12)

=-32-8-36

=-76

=2 =2(0-4+1(0-12+8(-1-6)

=2(-4) +1(-12) +8(-7)

=-8-12-56=76x= =2, y= = , z= x=2, y=2 and z=2.

Matrix inversion method:

(a). A= , X= and B=

AX=BX=A-1.BAnd A-1=

AdjA=

= = AdjA = DetA=

2(9)-4(-26) +5(1)

=136

A-1= X=A-1.B

=

=

==

x=3, y=1 and z=1.(b). Sol: let A= , X=

and B=

AX=BX=A-1.BAnd A-1=

=2

=2(1+1) +1(1-1) +3(-1-1)

=2(2) +1(0) +3(-2)

=4-6 = -2AdjA=

=


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Matrices

=

AdjA = A-1=

X=A-1.B=

=

== x=1, y=2 and z=3

(c). A= , X= and B=

=1 =1(-5-7) -1(-2-14) +1(2-10)

=1(-12) -1(-16) +1(-8)

=-12+16-8=- 4

AdjA=

= =

AdjA = A-1=

X=A-1.B=

=

==

x=1, y=3 and z=5.

(d).let A= , X= and B= =2 =2(-8-1) +1(4-3) +3(-1-6)

=2(-9) +1(1) +3(-7)

=--18+1-21

=-38 AdjA=

= = AdjA =

A-1= X=A-1.B

=

=

== x=2, y=2 and z=2.


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Matrices

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Matrices inter first year important