Trouble solving this Determinant

[shiri]

I'm having trouble solving this question:

If det

| a b c |
| p q r | = 2,
| u v w |

find det (3B^-1) where B =

| 2u 3u-a a-p |
| 2v 3v-b b-q |
| 2w 3w-c c-r |

Is it possible to transpose B?

So far what I got is 4; however, the instructor told me that the answer is |B| = 27/4

 

[LCKurtz]

Remember there is a difference between a determinant and a matrix. I'm guessing you are given a matrix

$$B =\left [<br /> \begin{array}{ccc}<br /> 2u&3u-a&a-p\\<br /> 2v&3v-b&b-q\\<br /> 2w&3w-c&c-r<br /> <br /> \end{array}<br /> \right ]$$

and what you have written is its determinant. Remember when you multiply a matrix by 3 it multiplies every element in the matrix. So for an n by n matrix A, if you multiply it by 3 and take its determinant, a 3 comes out of every column so

det(3A) = 3ⁿdet(A)

That plus the fact that det(A^-1) = 1 / det(A)
should help you out.