নির্ণায়ক পাওয়ার পয়েন্ট স্লাইড

1. wbY©vqK(𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒏𝒕)
‡h wbqgØviv cÖ‡Z¨KwUeM© g¨vwUª‡·i
GKwUgvb cvIqvhvq Zv‡K wbY©vqKe‡j|

2. wbY©vq‡Ki gvb wbY©qt
GK µ‡gi eM©g¨vwUª· 𝑨 = [𝒂 𝟏𝟏]
Gi wbY©vqK 𝑫 = 𝒂 𝟏𝟏 = 𝒂 𝟏𝟏
`yB µ‡gi eM©g¨vwUª·A=
𝑎11 𝑎12
𝑎21 𝑎22
Gi wbY©vqK 𝑫 =
𝑎11 𝑎12
𝑎21 𝑎22
= 𝒂 𝟏𝟏. 𝒂 𝟐𝟐 − 𝒂 𝟏𝟐. 𝒂 𝟐𝟏
[mvwi Kjv‡giAe¯’vb Abyhvqx fzw³¸wj ‡jLv|]

3. wbY©vq‡Ki gvbt
wZb µ‡gi eM© g¨vwUª·
A=
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
Gi wbY©vqK 𝑫 =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
= 𝒂 𝟏𝟏
𝑎22 𝑎23
𝑎32 𝑎33
− 𝒂 𝟏𝟐
𝑎21 𝑎23
𝑎31 𝑎33
+ 𝐚 𝟏𝟑
𝑎21 𝑎22
𝑎31 𝑎32
D`vniY:
2 1 1
0 5 −2
1 −3 4
Gigvb wbY©q Ki|D: 21

4. mvivR WvqvMÖv‡gi gva¨‡g wbY©vq‡Ki gvb †ei Kiv h
A=
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
=
𝑎 𝑏 𝑐 𝑎 𝑏
𝑑 𝑒 𝑓 𝑑 𝑒
𝑔 ℎ 𝑖 𝑔 ℎ
= 𝑎𝑒𝑖 + 𝑏𝑓𝑔 + 𝑐𝑑ℎ − 𝑔𝑒𝑐 − ℎ𝑓𝑎 − 𝑖𝑑𝑏
D`vniY:
2 1 −1
3 2 2
5 4 3
Gigvb wbY©q Ki|D: −5

5. wbY©vqK A=
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
fzw³ 𝑎 Gi Abyivwk
𝑒 𝑓
ℎ 𝑖
fzw³ 𝑏 Gi Abyivwk
𝑑 𝑓
𝑔 𝑖
fzw³ 𝑐 Gi Abyivwk
𝑑 𝑒
𝑔 ℎ
fzw³ 𝑑 Gi Abyivwk
𝑏 𝑐
ℎ 𝑖
fzw³ 𝑒 Gi Abyivwk
𝑎 𝑐
𝑔 𝑖
fzw³ 𝑔 Gi Abyivwk
𝑏 𝑐
𝑒 𝑓
BZ¨vw`
cÖ‡Z¨KwU fzw³i ¯^-¯^
mvwi- Kjvg eivei fzw³ ev`
†i‡LAewkó fzw³i Øviv
MwVZ wbY©vqKB
Abyivwk|
Abyivwk I mn¸YK

6. Abyivwk I mn¸YK
(𝒑, 𝒒)-Zg fzw³i wPý= −𝟏 𝒑+𝒒
Avi wPýhy³Abyivwk‡Kmn¸YK e‡j|
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
wbY©vq‡Kifzw³ 𝑎12 Gi
mn¸YK 𝐴12 = −1 1+2 𝑎21 𝑎23
𝑎31 𝑎33
= −
𝑎21 𝑎23
𝑎31 𝑎33
Abyiƒcfv‡e mKj fzw³i mn¸YK †ei Kiv hvq|

7. wbY©vqK A=
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
= 𝑎. −1 1+1 𝑒 𝑓
ℎ 𝑖
+ 𝑏. −1 1+2 𝑑 𝑓
𝑔 𝑖
+ 𝑐. −1 1+3 𝑑 𝑒
𝑔 ℎ
= 𝑎
𝑒 𝑓
ℎ 𝑖
− 𝑏
𝑑 𝑓
𝑔 𝑖
+ c
𝑑 𝑒
𝑔 ℎ
`ªóe¨:
1. wbY©vq‡Kigvb wbY©‡q 1g mvwi e¨envi Kiv n‡q‡Q|
2. ‡h †Kvbmvwi/Kjvg e¨envi K‡iGB GKB gvb cv‡ev|
mn¸Y‡Ki gva¨‡gwbY©vq‡Kigvb wbY©q

8. AbyivwkI mn¸YK
3 µ‡gi wbY©vq‡Kifyw³¸wjimn¸Y‡KiwPý wb¤œiƒct
−1 1+1 −1 1+2 −1 1+3
−1 2+1 −1 2+2 −1 2+3
−1 3+1 −1 3+2 −1 3+3
=
+ − +
− + −
+ − +
`ªóe¨t Avgiv mn¸Y‡Kifzw³¸wji wPý wn‡m‡eai‡Zcvwi|
(K) †Kvb wbY©vq‡Ki GKwU mvwi ev Kjv‡gifzw³¸wj‡K Zv‡`i
mn¸YKØviv ¸YK‡i†hvM Ki‡jwbY©vq‡Ki gvb cvIqv hvq|
(L) †KvbwbY©vq‡Ki GKwU mvwi ev Kjv‡gi fzw³¸wj‡KAb¨
mvwi ev Kjv‡gi fzw³i mn¸bKØviv ¸YK‡i †hvM Ki‡jwbY©vq‡Ki
gvb k~b¨nq|

9. (i) ‡KvbwbY©vq‡KiGKwU mvwi/Kjv‡gi mKj fzw³
k~b¨ n‡j wbY©vq‡Kigvb k~b¨ nq|
D`vt D=
𝑎 𝑏 0
𝑑 𝑒 0
𝑔 ℎ 0
= 0 BZ¨vw`|
(ii) ‡Kvb wbY©vq‡Kimvwi‡K Kjv‡g Ges Kjvg‡K
mvwi‡Z cwibZKi‡jZvi gv‡bi cwieZ©bnq bv|
D`vt
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
= D=
𝑎 𝑑 𝑔
𝑏 𝑒 ℎ
𝑐 𝑓 𝑖
wbY©vq‡Ki ag© (Properties of
determinants)

10. (iii) wbY©vq‡Ki `ywU mvwi/Kjvg ci¯úi ¯’vb wewbgq Ki‡jgvb
cwieZ©b nqbv Z‡e wPýcwieZ©b nq|
D`vt
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
= D,
𝑏 𝑎 𝑐
𝑒 𝑑 𝑓
ℎ 𝑔 𝑖
= −𝐷
(iv) ‡Kvb wbY©vq‡Ki `yBwU mvwi ev Kjvg GKB n‡jZvi gvb
ïb¨ nq|
D`vt D=
𝑎 𝑏 𝑏
𝑑 𝑒 𝑒
𝑔 ℎ ℎ
= 0
wbY©vq‡Kiag© (Properties of
determinants)

11. (v) wbY©vq‡KiGKwUmvwiev Kjvg‡K †KvbmsL¨vØviv ¸YKi‡j
Zvi gvb‡KH msL¨vØviv ¸YKi‡Zn‡e|
D`vt 𝐦
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
=
𝒎𝑎 𝒎𝑏 𝒎𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
= 𝒎D
(vi) ‡Kvb wbY©vq‡KiGKwUmvwiev KjvgGicÖ‡Z¨K fzw³‡K
`yBwUev Z‡ZvwaK msL¨vi†hvMdjiƒ‡ccÖKvk Kiv †M‡j
wbY©vqKwU‡KI`yB evZ‡ZvwaK wbY©vq‡Ki†hvMdjiƒ‡c cÖKvk
Kiv hv‡e|
D`vt
𝑎 + 𝑝 𝑏 𝑐
𝑑 + 𝑞 𝑒 𝑓
𝑔 + 𝑟 ℎ 𝑖
=
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
+
𝑝 𝑏 𝑐
𝑞 𝑒 𝑓
𝑟 ℎ 𝑖
wbY©vq‡Ki ag©

12. (vii) GKwUwbY©vq‡KiGKwUmvwievKjv‡gifzw³¸wjimv‡_H
wbY©vq‡KiAciGKev GKvwaKmvwiev Kjv‡gifzw³¸wji 𝑘 ¸wYZK
†hvM/we‡qvMKi‡jwbY©vq‡KigvbAcwiewZ©Z_v‡K|
D`vt
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
=
𝑎 + 𝑘𝑏 𝑏 𝑐
𝑑 + 𝑘𝑒 𝑒 𝑓
𝑔 + 𝑘ℎ ℎ 𝑖
= 𝐷
উদ াঃ ম ন ননর্ণয় করাঃ (মূলননয়মম ব ননর্ণ য়মকর ধমণ বযবহ র কমর)
13 16 19
14 17 20
15 18 21
wbY©vq‡Kiag© (Properties of determinants)

13. mvwii Bs‡iwR 𝑅𝑜𝑤 ms‡ÿ‡c 𝑅
Kjv‡gi Bs‡iwR 𝐶𝑜𝑙𝑢𝑚𝑛 ms‡ÿ‡c 𝐶
1g mvwi 𝑅1 ev 𝑟1
2q mvwi 𝑅2 ev 𝑟2
3q mvwi 𝑅3 ev 𝑟3
1g Kjvg 𝐶1 ev 𝑐1
2q Kjvg 𝐶2 ev 𝑐2
3q Kjvg 𝐶3 ev 𝑐3
`ªóe¨t †Kv‡bv wbY©vq‡Kimvwi 𝑅𝑜𝑤 I Kjvg
𝐶𝑜𝑙𝑢𝑚𝑛 Gi Kg© c×wZ ms‡ÿ‡c cÖKvk Kiv hvq|
‡hgbt1g mvwi †_‡K2q mvwi we‡qvMK‡i 1g mvwi‡Z
emv‡bv‡K 𝑅1
′
= 𝑅1 − 𝑅2 ev 𝑟1
′
= 𝑟1 − 𝑟2 Øviv
Ges 1g, 2q I 3q Kjvg †hvM K‡i Z…ZxqKjv‡g emv‡bv‡K
𝐶3
′
= 𝐶1 + 𝐶2 + 𝐶3 Øviv cÖKvk Kiv nq|
jÿYxqt

14. wbY©v‡qi gvb wbY©‡qi mvaviY Gj`wi`g/cÖwµqv
wbY©vq‡Ki fzw³mg~‡ni gvbhw` msL¨vq _v‡KZvn‡j mvaviYZmivmwi gvbwbY©q
c×wZ e¨enviKiv nq| wKš‘ wbY©vq‡Ki fzw³mg~‡ni gvbhw` exRMvwYwZK ivwk nq
Zvn‡j wb¤œwjwLZ avcmg~n AbymiY Kiv fv‡jv-
avc1: wbY©vq‡Ki ag© e¨envi K‡imvwi evKjv‡gi gvbmg~n‡K†QvUKivi †Póv Ki‡Z
n‡e|
avc2: wbY©vq‡Ki mvwi evKjv‡g †KvbmvaviY gvb_vK‡j ZvKgbwb‡Z n‡e|
avc3: †h †KvbGKwUmvwi evKjv‡gi fzw³‡K GKB Ki‡Zn‡e| Gi Rb¨ mvwii mv‡_mvwi
(Kjv‡gi mv‡_Kjvg)†hvM evwe‡qvM Kivhv‡e|
avc4: ‡h †Kvbmvwi evKjv‡gi `yBwU fzw³‡K k~b¨ Ki‡Z†Póv Ki‡Zn‡e| KviYk~‡b¨i
mv‡_ Zvimn¸Y‡Ki ¸Ydj k~b¨ nq|
avc5: †h mvwi evKjvgG `yBwU fzw³ k~b¨ _vK‡j †mB mvwi evKjvgGi gva¨‡gwe¯Ívi
Ki‡Zn‡e|
avc6: me©`v hvnv cÖgvYKi‡Z ejv n‡”Q†mB w`‡K †Lqvj †i‡L wbY©vq‡Ki Acv‡ikb¸wj
Pvwj‡q †h‡Z n‡e|
mKj‡Kab¨ev`