DETERMInants !!!!!!!!!!?

Question

please help got ma exams tomm


prove the following determinants r equal

ie prove tht


| (b+c-a) a a |
| b (c+a-b) b |
| (a+b-c) c c |






| a b c |
| b c a |
| c a b |

Answer 1

Just write out the determinants and simplify.

| a b c |
| b c a |
| c a b |

= a(bc - a²) - b(b² - ac) +c(ab - c²) = 3abc - a³ - b³ - c³.

| (b+c-a) a a |
| b (c+a-b) b |
| (a+b-c) c c |

= a(bc - (a + b - c)(c + a - b)) - b((b+c-a)c - a(a + b - c)) + c((b + c - a)(c + a - b) - ab)

= a(bc -(a² - b² -c² + 2bc)) - b(bc + c² - ac - a² - ab + ac) + c((c² - a² - b² + 2ab) - ab)

= - abc - a³ + ab² + ac² - b²c - bc² + a²b + ab² + c³ - a²c - b²c + abc

= - a³ + 2ab² + ac² - 2b²c - bc² + a²b + c³ - a²c.

As you can see, these determinants are not identical. Having rechecked my arithmetic, I can only conclude you've made an error in writing out the question.