若A爲三堦方陣,且|A 2E|=0,|2A E|=0,|3A-4E|=0,則|A|=?其中E爲單位陣.網友廻答：
- A=
a1,b1,c1
a2,b2,c2
a3,b3,c3
|A|=a1b2c3 a2b3c1 a3b1c2-a3b2c1-a2b1c3-a1b3c2
=a1(b2c3-b3c2) a2(b3c1-b1c3) a3(b1c2-b2c1)
|A 2E|=0=(a1 2)[(b2 2)(c3 2)-b3c2] a2[b3c1-b1(c3 2)] a3[b1c2-(b2 2)c1]
=a1(b2c3 2b2 2c3 2*2-b3c2) 2(b2c3 2b2 2c3 2*2-b3c2) a2(b3c1-b1c3)-2a2b1 a3(b1c2-b2c1)-2a3c1
=a1(b2c3-b3c2) a2(b3c1-b1c3) a3(b1c2-b2c1) 2a1(b2 c3) 2*2a1 2(b2c3-b3c2) 2*2(b2 c3) 2*2*2-2a2b1-2a3c1
=|A| 2*2*2 2*2(a1 b2 c3) 2(a1b2 a1c3 b2c3-b3c2-a2b1-a3c1)
=|A| 8 4(a1 b2 c3) 2(a1b2 a1c3 b2c3-b3c2-a2b1-a3c1)
|2A E|=0=2|A E/2|=2[|A| 1/(2*2*2) (a1 b2 c3)/(2*2) (a1b2 a1c3 b2c3-b3c2-a2b1-a3c1)/2]
=2|A| 1/4 (a1 b2 c3)/2 (a1b2 a1c3 b2c3-b3c2-a2b1-a3c1)
|A| 8 4(a1 b2 c3)=4|A| 1/2 (a1 b2 c3)
(a1 b2 c3)=|A|-5/2
(a1b2 a1c3 b2c3-b3c2-a2b1-a3c1)=-[2|A| 1/4 (a1 b2 c3)/2]=-5|A|/2 1
|3A-4E|=0=3|A-4E/3|=3[|A| (-4/3)^3 16(a1 b2 c3)/9-4(a1b2 a1c3 b2c3-b3c2-a2b1-a3c1)/3]
=3|A|-64/9 16|A|/3-40/3 10|A|-4=55|A|/3-220/9
|A|=4/3
網友廻答：
- 說明A的三個特征值分別是-2,-1/2,4/3.所以|A|=三個特征值相乘.
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