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I need to find the determinant of a matrix in $\Bbb F_5$.

I started with the following matrix

$\begin{bmatrix} 3 & 1 & 0 & 1\\1 & 2 & 1 & 0\\3 & 0 & 0 & 2\\1 & 2 & 3 & 4\end{bmatrix}$

and row reduced it

$\begin{bmatrix} 3 & 1 & 0 & 1\\0 & 4 & 0 & 1\\0 & 0 & 1 & 4\\0 & 0 & 0 & 1\end{bmatrix}$

In the process I swapped two rows, so I will need to multiply my determinant by $-1^1$.

Is it correct to suppose that I calculate the determinant by: 3 * 4 * 1 * 1 * $-1^1$ (mod 5)?

Extra question: It seems that for the process of row reduction I can use an arbitrary number of elementary row operations- so is it valid to further manipulate the matrix to an identity matrix and receive a determinant value which is a multiple of 1?

Please excuse my questions if they are silly or trivial, i do not have a strong background.

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    $\begingroup$ So, yes, the determinant is exactly as you describe. You could also do cofactor expansion. You could use more row operations to reduce the matrix to the identity (but remember to keep track of scaling factors too). $\endgroup$ – Michael Burr Dec 21 '17 at 20:52

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