Bounding the determinant of a matrix with bounded coefficients Suppose that $A$ is a real matrix of dimension $n \times n$ and that its coefficients are bounded by $c\ge0$ ($\vert a_{ij} \vert \le c$ for all $1\le i,j \le n$).
How to prove that
$$\vert \det A \vert \le c^n n^{n/2}$$

Since 

is ture for every element in the matrix, let us consider a matrix where all of the coefficients are the value of c. 
Now factor out the c throughout the entire matrix. 

Using a formula we obtain

Which reduces to

Using another formula

Reduces to

Since all elements in the matrix where of value c then all of the elements left are valued as 1. The result will be that the determinant of a matrix times a matrix transpose will be zero. 

