# Functions that generate “easy” matrices of full rank

While explaining how to invert matrices I once used this ill-fated example $A=\begin{pmatrix} 1&2&3\\4&5&6 \\7&8&9 \end{pmatrix}$ which can not be inverted ($\det(A)=0$). That got me thinking, given a matrix of size $N$, what are some good functions that map to the elements such that:

1. The elements are integers
2. The elements are "small" (for hand calculation)
3. The matrix is always invertible
4. (optional) the function has a random component, but still satisfies (3)

Let $A_{ij} = f(j + (i-1)N)$. In the example above $f(n) = n$.

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A random matrix is invertible with high probability, so just choose the entries randomly. Alternately, choose an upper-triangular matrix with nonzero diagonal entries. These are a little easier to invert than a general matrix but you'll still learn something from actually doing it. –  Qiaochu Yuan Aug 10 '12 at 1:03
@QiaochuYuan while that may be true, I want to be 100% positive that the rows are linearly independent. I guess I'm also interested in the problem itself, what functions generate matrices in $GL(\mathbb{Z},N)$ (did I say that correctly)? –  Hooked Aug 10 '12 at 1:04
The matrix with entries $a_{j,k}=\min(j,k)$ is symmetric and invertible. If you really want a guaranteed nonsingular matrix with integer entries, then multiply a lower triangular matrix and an upper triangular matrix, both having nonzero diagonal entries (as Qiaochu alludes to). –  Ｊ. Ｍ. Aug 10 '12 at 1:09
Have a look at the Pascal matrices as well. –  Ｊ. Ｍ. Aug 10 '12 at 1:20
You could generate polynomials with non-zero roots, and consider them the characteristic polynomial of a matrix. –  Arkamis Aug 10 '12 at 1:32
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There are many variants of this idea. For example, choose entries down the main diagonal not divisible by $3$, and the remaining entries in some row or some column divisible by $3$.