put the famous pascal triangle into a matrix. it could go intoa lower triangular $l$ or its transpose $l'$ or a symmetric matrix $s$: $$l=matrix{ [ 1 0 0 0 ] cr [ 1 1 1 1 ] cr [ 1 1 0 0 ] cr[ 1 2 1 0 ] cr[ 1 3 3 1 ] cr}quad l' =matrix{ [ 1 1 1 1]cr [ 0 1 2 3 ]cr [ 0 0 1 3 ] cr[ 0 0 0 1 ] cr}quad s = matrix{[ 1 2 3 4]cr [ 1 3 6 10]cr [ 1 4 10 20]cr}$$these binomial numbers come from a recursion, or from the formulafor $i$ choose $j$, or functionally from taking powers of $(1 + x)$. the amazing thing is that $l imes l' = s$. (ok for $4 imes 4$)it follows that $s$ has determinant 1. the matrices have otherunexpected properties too, that give beautiful examples in teachinglinear algebra. the proof of $l l' = s$ comes 3 ways: 1. by induction using the recursion formula for the matrix entries. 2. by an identity for the coefficients $i+j$ choose $j$ in $s$. 3. by applying both sides to the column vector $[ 1 x x^2 x^3 ... ]'$.the third way also gives a proof that $s^3 = -i$ but we doubt that result. the rows of the ``hypercube matrix" $l^2$ count corners and edgesand faces and ... in n dimensional cubes.