Determinant Prove

[nycmath]

See attachment.

[Eigenvalue]

A. (1-a³)-a(a²-a²)+a²(a⁴-a)
=1-a³+a⁶-a³
=a⁶-2a³+1
=(a³-1)²

B.[(ca)(a+b)-(ab)(c+a)]-bc(a+b-(c+a))+(b+c)(ab+ca)
=ca²+cab-a²b-cab-(bc)(b+c)+ab²+cab+cab+c²a
=ca²-a²b-b²c-c²b+a²b+2cab+c²a

(b-c)(c-a)(a-b)
=(bc-c²-ab-ac)(a-b)
=abc-ac²-a²b-a²c-b²c+bc²+ab²+acb
=2abc-ac²-a²b-a²c+ab²+bc²