(a) Let A = ⎣⎡2200−24121⎦⎤
So A²
= ⎣⎡2200−24121⎦⎤⎣⎡2200−24121⎦⎤
= ⎣⎡4+0+04−4+00+8+00+0+40+4+80−8+42+0+12−4+20+8+1⎦⎤ = ⎣⎡408412−4309⎦⎤
A³ = A²A
=⎣⎡408412−4309⎦⎤ ⎣⎡2200−24121⎦⎤
= ⎣⎡8+8+00+24+016−8+00−8+120−24+00+8+364+8+30+24+08−8+9⎦⎤
= ⎣⎡162484−244415249⎦⎤
Now A³ - A² =
⎣⎡162484−244415249⎦⎤ −⎣⎡408412−4309⎦⎤
= ⎣⎡122400−364812240⎦⎤
12A =
12⎣⎡2200−24121⎦⎤ =⎣⎡242400−2448122412⎦⎤
A³ - A² - 12A =
⎣⎡122400−364812240⎦⎤ −⎣⎡242400−2448122412⎦⎤
= ⎣⎡−12000−12000−12⎦⎤
= −12⎣⎡100010001⎦⎤=−12I
Relation is verified.
b)
A³ - A² - 12A =-12I
Multiplying both sides by A-1
A³A-1- A²A-1-12AA-1=-12IA-1
=> A²I - AI - 12I = -12A-1
=> A² - A - 12I = -12A-1
=> -12A-1 = ⎣⎡408412−4309⎦⎤ - ⎣⎡2200−24121⎦⎤ -⎣⎡120001200012⎦⎤
= ⎣⎡4−2−120−2−08−0−04−0−012+2−12−4−4−03−1−00−2−09−1−12⎦⎤ =⎣⎡−10−2842−82−2−4⎦⎤
A-1 = 61⎣⎡51−4−2−14−112⎦⎤
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