The value of the determinant `|(1,1,1),(.^(m)C_(1),.^(m +1)C_(1),.^(m+2)C_(1)),(.^(m)C_(2),.^(m +1)C_(2),.^(m+2)C_(2))|` is equal to

Options

A. 1
B. -1
C. 0
D. none of these

Answer

A

Explanation

`|{:(1,,1,,1),(.^(m)C_(1),,.^(m+1)C_(1),,.^(m+2)C_(1)),(.^(m)C_(2),,.^(m+1)C_(2),,.^(m+2)C_(2)):}|`
`=|{:(1,,1,,1),(.^(m)C_(1),,.^(m+1)C_(1),,.^(m+1)C_(0)+.^(m+1)C_(1)),(.^(m)C_(1),,.^(m+1)C_(2),,.^(m+1)C_(1)+.^(m+1)C_(2)):}|
|{:(1,,1,,1),(.^(m)C_(1),,.^(m+1)C_(1),,.^(m+1)C_(0)),(.^(m)C_(2),,,^(m+1)C_(2),,.^(m+1)C_(1)):}|" ""[Applying " C_(3) to C_(3) -C_(2)"]"`
`=|{:(1,,1,,1),(.^(m)C_(1),,.^(m)C_(0).^(m)C_(1),,.^(m+1)C_(0)),(.^(m)C_(2),,.^(m)C_(1)+.^(m)C_(2),,.^(m+1)C_(1)):}|`
`=|{:(1,,0,,0),(.^(m)C_(1),,.^(m)C_(0),,.^(m+1)C_(0)),(.^(m)C_(2),,.^(m)C_(1),,.^(m+1)C_(1)):}|" ""[Applying " C_(2) to C_(2) -C_(1)" ]"`
`=.^(m)C_(0).^(m+1)C_(1)-^(m+1)C_(0).^(m)C_(1)`
`=m +1-m`
`=1`