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ijàãîíàëüíà ìàòðèöÿ, â ÿêî¿ âñ³ åëåìåíòè ãîëîâíî¿ ä³àãîíàë³<br />
äîð³âíþþòü îäèíèö³, íàçèâàºòüñÿ îäèíè÷íîþ ³ ïîçíà÷à-<br />
ºòüñÿ áóêâîþ Å (I)<br />
⎛1 0... 0⎞<br />
⎜ ⎟<br />
0 1 ... 0<br />
E = ⎜ ⎟<br />
⎜ ... ... ... ... ⎟ .<br />
⎜<br />
00...1⎟<br />
⎝ ⎠<br />
2.2.3. ij¿ íàä ìàòðèöÿìè<br />
Ñóìîþ äâîõ ìàòðèöü îäíàêîâèõ ðîçì³ð³â íàçèâàºòüñÿ<br />
ìàòðèöÿ òîãî æ ðîçì³ðó, åëåìåíòè ÿêî¿ äîð³âíþþòü ñóìàì<br />
â³äïîâ³äíèõ åëåìåíò³â äîäàíê³â ìàòðèöü.<br />
Çà îçíà÷åííÿì ñóìà ìàòðèöü À ³ Â<br />
º ìàòðèöÿ<br />
⎛a11 a12 ... a1 n ⎞ ⎛b11 b12 ... b1<br />
n ⎞<br />
⎜ ⎟ ⎜ ⎟<br />
a21 a22 ... a2n<br />
b21 b22 ... b2<br />
n<br />
A= ⎜ ⎟ B = ⎜ ⎟<br />
⎜... ... ... ... ⎟ ⎜... ... ... ... ⎟<br />
⎜ a a ... a ⎟ ⎜ b b ... b ⎟<br />
⎝ m1 m2 mn⎠ ⎝ m1 m2<br />
mn⎠<br />
⎛a + b a + b ... a + b<br />
⎜<br />
⎜<br />
a + b a + b ... a + b<br />
11 11 12 12 1n<br />
1n<br />
21 21 22 22 2n<br />
2n<br />
= + = ⎜ ... ... ... ... ⎟ (2.2.1)<br />
⎜<br />
am 1<br />
+ bm 1<br />
am2 + bm 2...<br />
amn + b ⎟<br />
⎝ mn ⎠ .<br />
C A B<br />
Î÷åâèäíî, ùî À +  =  + À; À +0=0+À = À.<br />
Ìàòðèö³ À ³ Â íàçèâàþòüñÿ ïðîòèëåæíèìè, ÿêùî ¿õ ñóìà<br />
À + Â =0 º íóëü-ìàòðèöÿ. Ìàòðèöÿ, ïðîòèëåæíà ìàòðèö³ À,<br />
ïîçíà÷àºòüñÿ –À, ³ ¿¿ åëåìåíòè ïðîòèëåæí³ çà çíàêîì åëåìåíòàì<br />
ìàòðèö³ À. Òîä³ ð³çíèöÿ äâîõ ìàòðèöü À ³  ìîæå<br />
áóòè çàì³íåíà ñóìîþ À ³ (–Â):<br />
C = À – Â = À +(–Â).<br />
Äîáóòêîì ìàòðèö³ íà ÷èñëî (àáî ÷èñëà íà ìàòðèöþ) íàçèâàºòüñÿ<br />
ìàòðèöÿ, åëåìåíòè ÿêî¿ º äîáóòêè åëåìåíò³â äàíî¿<br />
ìàòðèö³ íà öå ÷èñëî<br />
⎞<br />
⎟<br />
⎟<br />
⎛a11 a12 ... a1 n ⎞ ⎛λa11 λa12 ... λa1<br />
n ⎞<br />
⎜ ⎟ ⎜ ⎟<br />
a21 a22 ... a2n<br />
λa21 λa22 ... λa2n<br />
λ A= Aλ = λ ⎜ ⎟ = ⎜ ⎟<br />
⎜... ... ... ... ⎟ ⎜ ... ... ... ... ⎟.<br />
⎜ a a ... a ⎟ ⎜ λa λa ... λa<br />
⎟<br />
⎝ m1 m2 mn ⎠ ⎝ m1 m2<br />
mn ⎠<br />
Î÷åâèäí³ âëàñòèâîñò³: λ (À + Â) =λÀ + λÂ; À⋅0 =0⋅À =0.<br />
Ïîð³âíþþ÷è âèçíà÷åííÿ îïåðàö³é äîäàâàííÿ, â³äí³ìàííÿ<br />
³ ìíîæåííÿ ìàòðèöü íà ÷èñëî ç àíàëîã³÷íèìè îïåðàö³ÿìè<br />
íàä âåêòîðàìè, áà÷èìî ¿õ àíàëîã³þ.<br />
Äîáóòêîì äâîõ ìàòðèöü À ³  (÷èñëî ñòîâïö³â ïåðøî¿<br />
ìàòðèö³ ïîâèííî äîð³âíþâàòè ÷èñëó ðÿäê³â äðóãî¿ ìàòðèö³)<br />
íàçèâàºòüñÿ òðåòÿ ìàòðèöÿ C, ó ÿê³é åëåìåíò c ij äîð³âíþº<br />
ñóì³ äîáóòê³â åëåìåíò³â i-ãî ðÿäêà ìàòðèö³ À íà â³äïîâ³äí³<br />
åëåìåíòè j-ãî ñòîâïöÿ ìàòðèö³ Â, òîáòî<br />
{ }<br />
1 1<br />
C = A⋅ B = c , äå c = a b + ... + a b , i = 1, m, j = 1, p . (2.2.2)<br />
ij ij i j in nj<br />
ßêùî ìàòðèöÿ À ìຠðîçì³ð m×n,  — n × p, òî Ñ áóäå<br />
ìàòè ðîçì³ð m × p.<br />
Äâ³ ìàòðèö³ íàçèâàþòüñÿ ïîãîäæåíèìè, ÿêùî ÷èñëî<br />
ñòîâïö³â ïåðøî¿ ìàòðèö³ äîð³âíþº ÷èñëó ðÿäê³â äðóãî¿.<br />
Ïðèêëàä 2.2.1. Çíàéòè äîáóòîê ìàòðèöü<br />
123 ⎛7⎞<br />
⎛ ⎞ ⎜ ⎟<br />
A= ⎜ ⎟ i B = 8<br />
456<br />
⎝ ⎠<br />
.<br />
⎜9<br />
⎟<br />
⎝ ⎠<br />
Ðîçâ’ÿçàííÿ. Ö³ ìàòðèö³ ïîãîäæåí³ (ê³ëüê³ñòü ñòîâïö³â<br />
ìàòðèö³ À äîð³âíþº 3, à ê³ëüê³ñòü ðÿäê³â ìàòðèö³  òåæ<br />
äîð³âíþº 3). Çà âèçíà÷åííÿì äîáóòêó çíàõîäèìî<br />
⎛7⎞<br />
⎛123⎞⎜ ⎟ ⎛17 ⋅ + 28 ⋅ + 39 ⋅ ⎞ ⎛50⎞<br />
AB ⋅ = ⎜ ⎟ 8<br />
456 = ⎜ ⎟=<br />
⎜ ⎟<br />
⎝ ⎠ ⎝47 ⋅ + 58 ⋅ + 69 ⋅ ⎠ ⎝122⎠<br />
.<br />
⎜9⎟<br />
⎝ ⎠<br />
46 47
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