若A为三阶方阵,且|A+2E|=0,|2A+E|=0,|3A-4E|=0,则|A|=?其中E为单位阵.

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

这应该算是高等代数的内容吧，

说明A的三个特征值分别是-2,-1/2,4/3.所以|A|=三个特征值相乘.