Question

Without expanding the determinants Prove that
`|{:(103,,115,,114),(111,,108,,106),(104,,113,,116):}| + |{:(113,,116,,104),(108,,106,,111),(115,,114,,103):}|=0`

Answer 1

We have `|{:(103,,115,,114),(111,,108,,106),(104,,113,,116):}| + |{:(113,,116,,104),(108,,106,,111),(115,,114,,103):}|`
We observe that the entries in determinants `Delta_(1) " and " Delta_(2)` are same (but not the same corresponding elements).
So let us simplify `Delta_(2)` in such a way that corresponding elements match with those of `Delta_(1)`
` Delta_(2)=|{:(113,,116,,104),(108,,106,,111),(115,,114,,103):}|`
` =-|{:(104,,116,,113),(111,,106,,108),(103,,114,,115):}|(c_(1) hArr C_(3))`
` =|{:(103,,114,,115),(111,,106,,108),(104,,116,,113):}|(R_(1) hArr R_(3))`
` =-|{:(103,,115,,114),(111,,108,,106),(104,,113,,116):}|(C_(1) hArr C_(3))`
`=- Delta_(1)`
`:. Delta_(1) + Delta_(2) =0`