Question

By using properties of determinants, prove the following: `|x+4 2x2x2xx+4 2x2x2xx+4|=(5x+4)(4-x)^2`

Answer 1

` Delta =|{:(x+4,,2x,,2x),(2x,,x+4,,2x),(2x,,2x,,x+4):}|`
Applying `R_(1) to R_(1) +R_(2) +R_(3),` we have
` Delta =|{:(5x+4,,5x+4,,5x+4),(2x,,x+4,,2x),(2x,,2x,,x+4):}|`
Applying `C_(2) to C_(2) -C_(1),C_(3) to C_(3)-C_(1)` we have
`Delta =(5x+4) (4-x)^(2) |{:(1,,0),(2x,,1),(,,):}|`
`=(5x+4) (4-x)^(2)`