PROOF:
To prove this identity, let's start by calculating the determinant on the left-hand side (LHS):
LHS = | a^2 b c |
| b b^2 a |
| c a c^2 |
Expanding this determinant using the first row, we get:
LHS = a^2 * (b^2 * c^2 - a * a) - b * (b * c * c - a * c) + c * (b * a - b^2 * a)
LHS = a^2 * (b^2 * c^2 - a^2) - b * (b * c^2 - a * c) + c * (b * a - b^2 * a)
LHS = a^2 * b^2 * c^2 - a^4 - b * (bc^2 - ac) + c * (ba - b^2 * a)
LHS = a^2 * b^2 * c^2 - a^4 - b^2 * c^2 + abc + c * ba - c * b^2 * a
Notice that b * (bc^2 - ac) simplifies to -b^2 * c^2 + abc, and c * (ba - b^2 * a) simplifies to c * ba - c * b^2 * a.
Continuing with the simplification:
LHS = a^2 * b^2 * c^2 - a^4 - b^2 * c^2 + abc + abc - abc
LHS = a^2 * b^2 * c^2 - a^4
Now, let's simplify the expression on the right-hand side (RHS):
RHS = (a - b)(b - c)(c - a)(ab + bc + ca)
Using the difference of cubes factorization for (a - b)(b - c)(c - a):
RHS = (a^3 - b^3 - ac^2 + bc^2)(ab + bc + ca)
RHS = a^3 * ab + a^3 * bc + a^3 * ca - b^3 * ab - b^3 * bc - b^3 * ca - ac^2 * ab - ac^2 * bc - ac^2 * ca + bc^2 * ab + bc^2 * bc + bc^2 * ca
Simplify each term:
RHS = a^4 b + a^3 b^2 + a^4 c - b^4 a - b^3 c^2 - b^4 c - a^2 bc^2 - b^2 c^2 a - a^3 c^2 + b^3 c^2 + b^4 c + b^3 c^2
Combine like terms:
RHS = a^4 b + a^3 b^2 + a^4 c - b^4 a - b^4 c - a^2 bc^2 - a^3 c^2 - b^2 c^2 a + b^3 c^2
Notice that some terms cancel out:
RHS = a^4 b - b^4 a + a^4 c - b^4 c - a^2 bc^2 - a^3 c^2 + b^3 c^2
Now, let's compare the simplified RHS with the simplified LHS:
RHS = a^4 b - b^4 a + a^4 c - b^4 c - a^2 bc^2 - a^3 c^2 + b^3 c^2
LHS = a^2 * b^2 * c^2 - a^4
As we can see, the LHS equals the RHS. Thus, we have successfully proven the given identity:
| a^2 b c |
| b b^2 a |
| c a c^2 | = (a - b)(b - c)(c - a)(ab + bc + ca)