Calculate the Determinant? $$D=\begin{bmatrix}
246 & 427 & 327 \\
1014 & 543 & 443 \\
-342 & 721 & 621 \\
\end{bmatrix}$$
What's the trick?
Hints?
Of course I know calculate by definition...
Please don't do that by cheating.
 A: To simplify the calculation, you can start by subtracting column three from column 2. That will not change the value of the determinant, but will simplify its calculation. Then subtract the first row from the other two, and you can then, essentially, compute the value of the determinant of a $2\times 2$ matrix by expansion of the minor using the appropriate column.
In problems like this (particularly when looking to simplify the calculation of determinants of even larger matrices, we can use elementary row/column operations can  to simplify the calculation of the determinant, but recall if and how each operation impacts the value of the original determinant.
A: First subtract the third column from the second as suggested by Gerry Myerson; this has no effect on the determinant.
$$\left[\begin{array}{ccc}
246 & 100 & 327\\
1014 & 100 & 443\\
-342 & 100 & 621
\end{array}\right]$$
Now divide the first column by $2$ and the second by $100$. These operations multiply the determinant by $\frac{1}{2}$ and $\frac{1}{100}$ respectively.
$$\left[\begin{array}{ccc}
123 & 1 & 327\\
507 & 1 & 443\\
-171 & 1 & 621
\end{array}\right]$$
Now add the first column to the third. This has no effect on the determinant.
$$\left[\begin{array}{ccc}
123 & 1 & 500\\
507 & 1 & 1000\\
-171 & 1 & 450
\end{array}\right]$$
Now divide the third column by $50$. This multiplies the determinant by $\frac{1}{50}$.
$$\left[\begin{array}{ccc}
123 & 1 & 10\\
507 & 1 & 20\\
-171 & 1 & 9
\end{array}\right]$$
Now substract the third row from the first and second. These have no effect on the determinant.
$$\left[\begin{array}{ccc}
294 & 0 & 1\\
681 & 0 & 11\\
-171 & 1 & 9
\end{array}\right]$$
Now divide the first column by $3$. This multiplies the determinant by $\frac{1}{3}$.
$$\left[\begin{array}{ccc}
98 & 0 & 1\\
227 & 0 & 11\\
-57 & 1 & 9
\end{array}\right]$$
If you expand along the second column, you should find that the computations are simple enough. To obtain the determinant of the original matrix, undo all of the effects of the column operations.
A: Thanks, everyone, after check the solution of the book(have a glance). I found that there is still a trick after the collect of 100. 
I calculate 3 times to come across this simple answer. 
\begin{align*}
d &= \left|\begin{array}{ccc} 246 & 427 & 327 \\ 1014 & 543 & 443 \\ -342 & 721 & 621 \\\end{array}\right| \\
&= \left|\begin{array}{ccc} 246 & 100 & 327 \\ 1014 & 100 & 443 \\ -342 & 100 & 621 \\\end{array}\right| \\
&= \left|\begin{array}{ccc} 246 & 100 & 327 \\ 768 & 0 & 116 \\ -588 & 0 & 294 \\\end{array}\right| \\
&=200 \left|\begin{array}{ccc} 123 & 1 & 327 \\ 384 & 0 & 116 \\ -294 & 0 & 294 \\\end{array}\right| \\
&=200 \left|\begin{array}{ccc} 123 & 1 & 450 \\ 384 & 0 & 500 \\ -294 & 0 & 0 \\\end{array}\right| \\
&=200 \times -294\times 500 \\
&=-29400000
\end{align*}
A: Sarrus rule:
$$\det(A) = \begin{vmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{vmatrix} 
= a_{11}a_{22}a_{33} + a_{21}a_{32}a_{13} + a_{31}a_{12}a_{23}
-a_{11}a_{32}a_{23} - a_{21}a_{12}a_{33} - a_{31}a_{22}a_{13}$$
Look for example here: http://www.math.utah.edu/~gustafso/determinants.pdf
