साबित करे कि
`|{:(alpha,beta,gamma),(alpha^(2),beta^(2),gamma^(2)),(beta+gamma,gamma+alpha,alpha+beta):}|=(beta-gamma)(gamma-alpha)(alpha-beta)(alpha+beta+gamma)`.

Explanation

माना कि `Delta=|{:(alpha,beta,gamma),(alpha^(2),beta^(2),gamma^(2)),(beta+alpha,gamma+alpha,alpha+beta):}|`
`=|{:(alpha,beta,gamma),(alpha^(2),beta^(2),gamma^(2)),(alpha+beta+gamma,alpha+beta+gamma,alpha+beta+gamma):}|=(alpha+beta+gamma)|{:(alpha,beta,gamma),(alpha^(2),beta^(2),gamma^(2)),(1,1,1):}|" "[R_(3)" से "(alpha+beta+gamma)" common लेने पर"]`
`=(alpha+beta+gamma)|{:(alpha-beta,beta-gamma,gamma),(alpha^(2)-beta^(2),beta^(2)-gamma^(2),gamma^(2)),(0,0,1):}|" "{:[(C_(1)toC_(1)-C_(2)),(C_(2)toC_(2)-C_(3))]:}`
`=(alpha+beta+gamma)(alpha-beta)(beta-gamma)|{:(1,1,gamma),(alpha+beta,beta+gamma,gamma^(2)),(0,0,1):}|" "[C_(1)" से "(alpha-beta)"common तथा "C_(2)"से "(beta-gamma)"common लेने पर"]`
`=(alpha+beta+gamma)(alpha-beta)(beta-gamma)*1*|{:(1,1),(alpha+beta,beta+gamma):}|" "[R_(3)" के अनुदिश विस्तार करने पर "]`
`(alpha+beta+gamma)(alpha-beta)(beta+gamma-alpha-beta)=(alpha+beta+gamma)(alpha-beta)(beta-gamma)(gamma-alpha)`