If `A=[(a, b, c), (x, y, z), (p, q, r)], B=[(q,-b, y),(-p, a,-x),(r,-c, z)]` and if `A` is invertible, then which of the following is not true? (a) `|A|=|B|` (b) `|A|=-|B|` (c) `|adj A|=|adj B|` (d) `A` is invertible if and only if `B` is invertible

Options

A. `|A|=|B|`
B. `|A|=-|B|`
C. `|"adj A"|=|"adj B"|`
D. A is invertible if and only if B is invertible

Answer

A

Explanation

`|B|=|(q,-b,y),(-p,a,-x),(r,-c,z)|` (Multiplying `R_(2)` by -1)
`=-|(q,-b,y),(p,-a,x),(r,-c,z)|` (Multiplying `C_(2)` nu -1)
`=|(q,b,y),(p,a,x),(r,c,z)|` (Changing `R_(1)` with `R_(2)`)
`=-|(p,a,x),(q,b,y),(r,c,z)|`
`=- |(a,x,p),(b,y,q),(c,z,r)|`
Hence, `|A|=-|B|`, obviously when `|A| ne 0, |B| ne 0`.
Also, `|"adj. B"|=|B|^(2)=(-|A|)^(2)=|A|^(2)`.