Given that the 3 x3 matrix:
|a b c|
|d e f|=6
|g h i|
find the 3x3 matrix:
|-3b -3a -3c/2|
| e d f/2|
| h-4e g-4d (i-4f)/2|
Any clue how to begin this question?
Use elementary matrices for example first define
$\displaystyle A=\begin{bmatrix}a & b & c \\ d & e &f \\ g & h& i \end{bmatrix}$
Then use
$\displaystyle E_1=\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 &0 \\ 0 & 0& 1 \end{bmatrix}$
Multiplying by this matrix has the effect of swapping column 1 and column 2
$\displaystyle AE_1=\begin{bmatrix}b & a & c \\ e & d &f \\ h & g& i \end{bmatrix}$
Now just come up with the other elementary matrices to do the rest of your row or column operations and use the fact that
$\displaystyle \det(AB)=\det(A)\cdot \det(B)$
TheEmptySet is right; basically, you are applying elementary row (or column) operations to the first matrix to get the second one. Each row operation does something specific to the determinant.
I just wanted to ask something that wasn't clear to me, from the way that you asked the question...
You ARE aware that they are asking for determinants, correct? Just asking because I didn't see you use the word anywhere, but you did say stuff like "Matrix A = 6", which doesn't really make any sense.
Do another elementary row operation: row swapping, adding or subtracting rows, or multiplying by a scalar. Keep doing that until you obtain the 2nd matrix.
Also note, adding/subtracting rows has no affect on the determinant; multiplication scales the determinant by the factor of x; and swapping multiplies the det by -1.
Check Property No 5