which of the following is`//`are true for
`Delta= |{:(a^(2),,1,,a+c),(0,,b^(2)+1,,b+c),(0,,b+c,,c^(2)+1):}|` ?

Options

A. `Delta ge 0` for real values of a,b,c
B. `Delta le 0` for real values of a,b,c
C. `Delta =|{:(bc-1,,0,,0),(1,,ac,,-a),(-b,,-a,,ab):}|`
D. `Delta =0` if bc =1 where a,b,c are non-zero

Answer

A, C, D

Explanation

`|{:(a^(2),,1,,a+c),(0,,b^(2)+1,,b+c),(0,,b+c,,c^(1)+1):}|=|{:(a,,0,,1),(0,,b,,1),(0,,1,,c):}||{:(a,,0,,1),(0,,b,,1),(0,,1,,c):}|`
`=|{:(a,,0,,1),(0,,b,,1),(0,,1,,c):}|^(2)`
`=(abc -a)^(2)= a^(2) (bc-1)^(2)`
`=|{:(bc-1,,0,,0),(1,,ac,,-a),(-b,,-a,,ab):}|`
{Determinant formed by cofactors)