AwZmswÿß cÖkœt

1. GKK ev A‡f` g¨vwUª· (Unit Matrix) Kv‡K e‡j?

DËit ‡h eM©Kvi g¨vwU ª‡·i cÖavb K‡Y©i mKj Dcv`vb GKK Ges evKx Dcv`vb k~b¨ Zv‡K GKK g¨vwUª· e‡j|

‡hgbt 1 0 00 1 00 0 1

2. k~b¨ g¨vwUª· (Null Matrix) Kv‡K e‡j?

DËit ‡Kvb g¨vwU ª‡·i mKj Dcv`vb k~b¨ n‡j Zv‡K k~b¨ g¨vwUª· e‡j|

‡hgbt 0 0 00 0 00 0 0

3. wecixZ g¨vwUª· (Inverse Matrix) Kv‡K e‡j?

DËit ywU eM©Kvi g¨vwUª‡·i ¸bdj GKwU GKK †gwUª· n‡j GKwU‡K AciwUi wecixZ (Inverse

Matrix) g¨vwUª· e‡j|

4.wm½yjvi †gwUª· Kv‡K e‡j?

DËit ‡Kvb eM©vKvi g¨vwUª· Gi Dcv`vb Øviv MwVZ wbb©vq‡Ki gvb k~b¨ n‡j H g¨vwU ª·‡K wm½yjvi g¨vwUª· e‡j|

5.wm½yjvi g¨vwUª· Gi ywU •ewkó¨ wjL?

DËit (i)hw` A GKwU wm½yjvi g¨vwUª· nq, Z‡e A = 0 A_v©r wbY©vq‡Ki gvb k~b¨ n‡e|

(ii)GwU GKwU eM©Kvi g¨vwUª· n‡e|

6.AbyeÜx g¨vwUª· (Adjoint Matrix) Kv‡K e‡j?

DËit GKwU eM©Kvi g¨vwUª‡·i Dcv`v‡bi mn¸bK Øviv MwVZ g¨vwUª·i cvk¦©Pi g¨vwUª·‡K AbyeÜx g¨vwUª· (Adjoint

Matrix) e‡j|

7.cvk©¦Pi g¨vwUª· (Transpose Matrix) Kv‡K e‡j?

DËit ‡Kvb g¨vwUª‡·i mvwi‡K Kjv‡g Ges Kjvg‡K mvwi‡Z e`j K‡i †h g¨vwUª· cvIqv hvq Zv‡K cvk©¦Pi

g¨vwUª· e‡j|

8.cÖwZmg g¨vwUª· (Symmetrix Matrix) Kv‡K e‡j?

DËit hw` †Kvb eM©Kvi g¨vwUª· Zvi cvk¦©Pi g¨vwUª‡·i mgvb nq Z‡e Zv‡K cÖwZmg g¨vwUª· e‡j|

9.g¨vwUª· Gi gvbv¼ (Rank of Matrix) Kv‡K e‡j?

DËit †Kvb g¨vwUª‡·i me©e„nr Ak~b¨K eM© Abyivwk wbY©vq‡Ki µg‡K H g¨vwUª‡·i gvbv¼ e‡j|

10.𝟑𝟕𝟗

Gi AWv©i ev gvÎv KZ?

DËit379

Gi AWv©i ev gvÎv = 3 × 1

11. 𝟐 𝟑 𝟒 Gi AWv©i ev gvÎv KZ?

DËit 2 3 4 Gi AWv©i ev gvÎv = 1 × 3

12.1 42 53 6

Gi cvk©¦Pi g¨vwUª· (Transpose Matrix) wjL|

DËit1 42 53 6

Gi cvk©¦Pi g¨vwUª· (Transpose Matrix) =1 2 34 5 6

(Ans)

13.𝐀 =𝐩 𝟐 𝟑𝐩 𝐱 − 𝟐 𝟔

Ges 𝐁 =𝐩 𝟐 𝐲 + 𝟏𝐩 𝟑 𝟔

Ges 𝐀 = 𝐁 nq, Z‡e 𝐱 I 𝐲 Gi gvb KZ?

mgvavbt ‡h‡nZz,A = B †m‡nZz G‡`i Abyiƒc Dcv`vb¸‡jv ci¯úi mgvb n‡e|

AZGe, x − 2 = 3 Ges y + 1 = 3

∴ x = 5 ∴ y = 2

∴ wb‡Y©q mgvavb t (x, y) = (5,1)

14.𝐀 =𝟏 𝟎 𝟏𝟒 𝟏 𝟎

Ges 𝐁 = 1 1 2 −1−1 2

nq, Z‡e 𝐀 × 𝐁 Gi gvb KZ?

mgvavbt A × B = AB=1 0 14 1 0

1 1 2 −1−1 2

= 1.1 + 0.2 + 1 −1 1.1 + 0 −1 + 1.24.1 + 1.2 + 0(−1) 4.1 + 1 −1 + 0.2

= 1 + 0 − 1 1 + 0 + 24 + 2 + 0 4 − 1 + 0

=0 36 3

(Ans)

15.𝐀 =5 71 −1

Ges 𝐁 =𝟏𝟐

nq, Z‡e 𝐀 × 𝐁 Gi gvb KZ?

mgvavbt A × B = AB =5 71 −1

12

=5.1 + 7.2

1.1 + (−1)2 =

5 + 141 − 2

=19−1

(Ans)

mswÿß I iPbvg~jK cÖkœt

1. 𝐀 = 2 3 4 1 2 3−1 1 2

Ges 𝐁 = 1 3 0−1 2 1 0 0 2

nq, Z‡e 𝐀 + 𝐁, 𝐀 − 𝐁 Ges 𝐀 × 𝐁 Gi gvb KZ?

mgvavbt ‡`Iqv Av‡Q, A =2 3 41 2 3

−1 1 2 Ges B =

1 3 0−1 2 1 0 0 2

GLb,A + B = 2 3 4 1 2 3−1 1 2

+ 1 3 0−1 2 1 0 0 2

=2 + 1 3 + 3 4 + 01 − 1 2 + 2 3 + 1

−1 + 0 1 + 0 2 + 2=

3 6 4 0 4 4−1 1 4

(Ans)

Avevi,A − B = 2 3 4 1 2 3−1 1 2

− 1 3 0−1 2 1 0 0 2

=2 − 1 3 − 3 4 − 01 + 1 2 − 2 3 − 1

−1 − 0 1 − 0 2 − 2=

1 0 4 2 0 2−1 1 0

(Ans)

Avevi,A × B = 2 3 4 1 2 3−1 1 2

1 3 0−1 2 1 0 0 2

=

2.1 + 3 −1 + 4.0 2.3 + 3.2 + 4.0 2.0 + 3.1 + 4.21.1 + 2 −1 + 3.0 1.3 + 2.2 + 3.0 1.0 + 2.1 + 3.2

−1.1 + 1 −1 + 2.0 −1.3 + 1.2 + 2.0 −1.0 + 1.1 + 2.2

= 2 − 3 6 + 6 3 + 8 1 − 2 3 + 4 2 + 6−1 − 1 −3 + 2 1 + 4

= −1 12 11 −1 7 8−2 −1 5

(Ans)

2. 𝐀 =1 2 31 3 41 4 3

nq, Z‡e 𝐀 Gi AbyeÜx Ges wecixZ g¨vwUª· wbY©q Ki|

mgvavbt ‡`Iqv Av‡Q, 𝐀 =1 2 31 3 41 4 3

GLb, A =1 2 31 3 41 4 3

= 1 9 − 16 − 2 3 − 4 + 3 4 − 3 = −7 + 2 + 3 = −2

A Gi Dcv`vb¸‡jvi mn¸bK¸‡jv-

𝐴11 =3 44 3

= 9 − 16 = −7

𝐴12 = −1 41 3

= −(3 − 4) = 1

𝐴13 =1 31 4

= 4 − 3 = 1

𝐴21 = −2 34 3

= −(6 − 12) = 6

𝐴22 =1 31 3

= 3 − 3 = 0

𝐴23 = −1 21 4

= − 4 − 2 = −2

𝐴31 =2 33 4

= 8 − 9 = −1

𝐴32 = −1 31 4

= − 4 − 3 = −1

𝐴33 =1 21 3

= 3 − 2 = 1

GLb A Gi Dcv`vb¸‡jvi mn¸bK w`‡q MwVZ g¨vwUª·- B =−7 1 1 6 0 −2−1 −1 1

∴ AbyeÜx g¨vwUª· AdjA = BT = −7 6 −1 1 0 −1 1 −2 1

(Ans)

Avgiv Rvwb, wecixZ g¨vwUª· A−1 =AdjA

A=

1

−2

−7 6 −1 1 0 −1 1 −2 1

(Ans)

3. 𝐀 =1 0 22 1 03 2 1

nq, Z‡e 𝐀 Gi AbyeÜx Ges wecixZ g¨vwUª· wbY©q Ki|

mgvavbt ‡`Iqv Av‡Q, 𝐀 =1 0 22 1 03 2 1

GLb, A =1 0 22 1 03 2 1

= 1 1 − 0 − 0 + 2 4 − 3

= 1 − 0 + 2 = 3

A Gi Dcv`vb¸‡jvi mn¸bK¸‡jv-

A11 =

1 02 1

= 1 − 0 = 1

A12 = −2 03 1

= − 2 − 0 = −2

A13 =2 13 2

= 4 − 3 = 1

A21 = −0 22 1

= −(0 − 4) = 4

A22 =1 23 1

= 1 − 6 = −5

A23 = −1 03 2

= − 2 − 0 = −2

A31 =0 21 0

= 0 − 2 = −2

A32 = −1 22 0

= − 0 − 4 = 4

A33 =1 02 1

= 1 − 0 = 1

GLb A Gi Dcv`vb¸‡jvi mn¸bK w`‡q MwVZ g¨vwUª·-

B = 1 −2 1 4 −5 −2−2 4 1

∴ AbyeÜx g¨vwUª· AdjA = BT = 1 4 −2 −2 −5 4 1 −2 1

(Ans)

Avgiv Rvwb, wecixZ g¨vwUª· A−1 =AdjA

A=

1

3

1 4 −2 −2 −5 4 1 −2 1

(Ans)

4. 𝐀 =1 1 12 2 31 4 9

nq, Z‡e 𝐀 Gi cvk¦©Pi, AbyeÜx Ges wecixZ g¨vwUª· wbY©q Ki|

mgvavbt ‡`Iqv Av‡Q, 𝐀 =1 1 12 2 31 4 9

, 𝐀 Gicvk¦©Pi g¨vwUª·, 𝐀𝐓 =1 2 11 2 41 3 9

(Ans)

GLb, A =1 1 12 2 31 4 9

= 1 18 − 12 − 1 18 − 3 + 1 8 − 2 = 6 − 15 + 6 = −3

A Gi Dcv`vb¸‡jvi mn¸bK¸‡jv-

A11 =2 34 9

= 18 − 12 = 6

A12 = −2 31 9

= − 18 − 3 = −15

A13 =2 21 4

= 8 − 2 = 6

A21 = −1 14 9

= − 9 − 4 = −5

A22 =1 11 9

= 9 − 1 = 8

A23 = −1 11 4

= − 4 − 1 = −3

A31 =1 12 3

= 3 − 2 = 1

A32 = −1 12 3

= − 3 − 2 = −1

A33 =1 12 2

= 2 − 2 = 0

GLb A Gi Dcv`vb¸‡jvi mn¸bK w`‡q MwVZ g¨vwUª·- B = 6 −15 6−5 8 −3 1 −1 0

∴ AbyeÜx g¨vwUª· AdjA = BT = 6 −5 1

−15 8 −1 6 −3 0

(Ans)

Avgiv Rvwb, wecixZ g¨vwUª· A−1 =AdjA

A=

1

−3

6 −5 1−15 8 −1 6 −3 0

(Ans)

5.𝐀 =𝟏 𝟐 𝟑𝟏 𝟐 𝟓𝟐 𝟒 𝟖

g¨vwU·wUi gvbv¼ ev gvÎv (Rank) wbY©q Ki|

mgvavbt ‡`Iqv Av‡Q, A =1 2 31 2 52 4 8

GLb, A =1 2 31 2 52 4 8

= 1 16 − 20 − 2 8 − 10 + 3 4 − 4 = 1 −4 − 2 −2 +

0 = −4 + 4 = 0

‡h‡nZz, A = 0 myZivs A g¨vwU·wUi gvbv¼ ev gvÎv (Rank) ≠3 A_v©r 3 Gi †P‡q †QvU n‡e|

GLb, 2 × 2 AW©v‡ii wbY©vqK wb‡q cvB,

A11 =2 54 8

= 16 − 20 = −4

‡h‡nZz, 2 × 2 AW©v‡ii wbY©vq‡Ki gvb = −4 ≠ 0 (Ak~b¨),

myZivs A g¨vwU·wUi gvbv¼ ev gvÎv (Rank) =2 (Ans)

06.𝐀 =𝟏 𝟐 𝟑𝟐 𝟑 𝟓𝟏 𝟑 𝟒

𝟐𝟏𝟓

g¨vwU·wUi gvbv¼ ev gvÎv (Rank) wbY©q Ki|

mgvavbt ‡`Iqv Av‡Q, A =1 2 32 3 51 3 4

215

GLb,A Gi Dcv`vb Øviv MwVZ 3 × 3 AW©v‡ii wbY©vqK -

GLb, A1 =1 2 32 3 51 3 4

= 1 12 − 15 − 2 8 − 5 + 3 6 − 3 = −3 − 6 + 9 = 0

Avevi, A2 =1 2 22 3 11 3 5

= 1 15 − 3 − 2 10 − 1 + 2 6 − 3 = 12 − 18 + 6 = 0

Avevi, A3 =1 3 22 5 11 4 5

= 1 25 − 4 − 3 10 − 1 + 2 8 − 5 = 21 − 27 + 6 = 0

Avevi, A4 =2 3 23 5 13 4 5

= 2 25 − 4 − 3 15 − 3 + 2 12 − 15 = 42 − 36 − 6 = 0

‡h‡nZz,3 × 3 AW©v‡ii mKj wbY©vqK gvb k~b¨ myZivs A g¨vwU·wUi gvbv¼ ev gvÎv (Rank) ≠3 A_v©r

3 Gi †P‡q †QvU n‡e|

GLb, 2 × 2 AW©v‡ii wbY©vqK wb‡Z n‡e, ∴ A1 =1 22 3

= 3 − 4 = −1

‡h‡nZz, 2 × 2 AW©v‡ii wbY©vq‡Ki gvb = −1 ≠ 0 (Ak~b¨), myZivs A g¨vwU·wUi gvbv¼ ev gvÎv

(Rank) =2 (Ans)

07.g¨vwUª‡·i mvnv‡h¨ mgvavb wbb©q Ki|

x+y+z = 6

5x-y+2z = 9

3x+6y−5z = 0

mgvavbt g¨vwU ª‡·i mvnv‡h¨ cÖ`Ë mgxKib wZbwU‡K †jLv hvq, AX=L ∴ X=A-1L………….(i)

‡hLv‡b, A =1 1 15 −1 23 6 −5

, X =xyz

, L =690

GLb, A =1 1 15 −1 23 6 −5

= 1 5 − 12 − 1 −25 − 6 + 1 30 + 3 = −7 + 31 + 33 = 57

A Gi Dcv`vb¸‡jvi mn¸bK¸‡jv-

A11 =−1 26 −5

= 5 − 12 = −7

A12 = −5 23 −5

= − −25 − 6 = 31

A13 =5 −13 6

= 30 + 3 = 33

A21 = −1 16 −5

= − −5 − 6 = 11

A22 =1 13 −5

= −5 − 3 = −8

A23 = −1 13 6

= − 6 − 3 = −3

A31 = 1 1−1 2

= 2 + 1 = 3

A32 = −1 15 2

= − 2 − 5 = 3

A33 =1 15 −1

= −1 − 5 = −6

GLb A Gi Dcv`vb¸‡jvi mn¸bK w`‡q MwVZ g¨vwUª·- B = −7 31 3311 −8 −33 3 −6

∴ AdjA = BT = −7 11 3 31 −8 333 −3 −6

∴ A−1=AdjA

A =

1

57

−7 11 3 31 −8 333 −3 −6

(i)bs mgxKib nB‡Z cvB, X= A-1L

xyz

=1

57

−7 11 3 31 −8 333 −3 −6

690

=1

57

−42 + 99 + 0 186 − 72 + 0 198 − 27 + 0

=1

57

57114171

=123

∴ wb‡b©q mgvavbt x = 𝟏, y = 2, z = 3