# Math Help - Determinant Question

1. ## Determinant Question

Let A = $
\begin{array}{c}\ \\ \\ \end{array}\;\begin{vmatrix}\;a & b & c \;\\\;p & q & q \;\\\;u & v & w\end{vmatrix}
$

and assume that det A = 3.

Compute:

det(2c^-1) where C = $
\begin{array}{c}\ \\ \\ \end{array}\;\begin{vmatrix}\;2p & -a+u & 3u \;\\\;2q & -b+v & 3v \;\\\;2r & -c+w & 3w\end{vmatrix}
$

I'm getting an answer of 1/3, which is wrong. The answer is 4/9.

The troubles I'm having you can read about right HERE. I know how to do everything, but I'm obviously just making a small mistake.

One specific question I would like to ask, is how do I remove the 2 from inside the det() brackets? I haven't been shown an example of that, and can't find one.

Thanks!

2. Originally Posted by Thomas
Let A = $
\begin{array}{c}\ \\ \\ \end{array}\;\begin{vmatrix}\;a & b & c \;\\\;p & q & q \;\\\;u & v & w\end{vmatrix}
$

and assume that det A = 3.

Compute:

det(2c^-1) where C = $
\begin{array}{c}\ \\ \\ \end{array}\;\begin{vmatrix}\;2p & -a+u & 3u \;\\\;2q & -b+v & 3v \;\\\;2r & -c+w & 3w\end{vmatrix}
$

I'm getting an answer of 1/3, which is wrong. The answer is 9/4.

The troubles I'm having you can read about right HERE. I know how to do everything, but I'm obviously just making a small mistake.

One specific question I would like to ask, is how do I remove the 2 from inside the det() brackets? I haven't been shown an example of that, and can't find one.

Thanks!
actually, for any real number c, det(cA) = c^n det(A) where n is the size of the matrix. i hope, this would help..

3. Originally Posted by kalagota
actually, for any real number c, det(cA) = c^n det(A) where n is the size of the matrix. i hope, this would help..
.. hmm, i get a 4/9.. just try to solve it.. maybe there was a mistake on my computation.. Ü

4. Oops, sorry. The answer is 4/9. I just mixed up the numbers.

Could you explain how you did it?

5. Originally Posted by Thomas
Let A = $
\begin{array}{c}\ \\ \\ \end{array}\;\begin{vmatrix}\;a & b & c \;\\\;p & q & q \;\\\;u & v & w\end{vmatrix}
$

and assume that det A = 3.

Compute:

det(2c^-1) where C = $
\begin{array}{c}\ \\ \\ \end{array}\;\begin{vmatrix}\;2p & -a+u & 3u \;\\\;2q & -b+v & 3v \;\\\;2r & -c+w & 3w\end{vmatrix}
$

I'm getting an answer of 1/3, which is wrong. The answer is 4/9.

The troubles I'm having you can read about right HERE. I know how to do everything, but I'm obviously just making a small mistake.

One specific question I would like to ask, is how do I remove the 2 from inside the det() brackets? I haven't been shown an example of that, and can't find one.

Thanks!
ok.. let's recall some theorems..

THEOREMS
Let A = [A_1 A_2 ... A_n] be an nxn matrix, where A_i are the columns of A for i=1,..,n.
let B be the transpose of A.
1) det (A) = det (B)
2) det([A_1 ... cA_k ... A_n]) = c det([A_1 ... A_k ... A_n])
3) det(cA) = det(c[A_1 A_2 ... A_n] = (c^n) det(A)
4) det([A_1 ... (A_k)+(A_j) ... A_n]) = det([A_1 ... A_k ... A_n]) + det([A_1 ... A_j ... A_n])
5) det([A_1 ... A_k cA_j ... A_n]) = 0 if A_k = A_j
6) det([A_1 ... A_k ... A_j .... A_n]) = (-1)^p det([A_1 ... A_j ... A_k .... A_n]) where p= j-k