For the lack of a better term, I will call the 2x2 matrices used to describe Primitive Pythagorean Triples in my last blog as "the matrices".
One of the consequences of the way they are constructed is that the two numbers in the right column represent the difference and sum of the two numbers in the left column.
This allows you to construct the full 2x2 matrix when any two of the elements are known. This, in conjunction with the knowledge that each offspring on the Barning tree has an incenter that was one of the three excenters of the parent, allow us to produce the tree without the use of matrix transformations.
If we start with the 3,4,5 triangle, which has a matrix ) we have already established that the products of the bottom row and the two diagonals give the radii of the three excircles (1x3=3, 1x2=2, 2x3=6)... So we can find the three offspring of the 3,4,5 triangle by promoting each of these excircles to an incircle, as shown here:
Now by using the fact that the right hand column is made up of the difference and sum of the left hand values, we can complete the two missing values to find the descendants of the parent matrix.
But we don't really want to know the matrices, we want the Pythagorean triples. Fortunately it is easy to find the triple associated with any of the matrices. Multiply down the left column and double to get the even leg; multiply down the right column to get the odd leg, and then sum the products of the two diagonals to get the hypotenuse. So the matrix on the lower left with rows of 2,1 and 3,5 will have an even leg that is twice 2x3, or 12. Its odd leg is 1x5 = 5 and the hypotenuse is 1x5 + 1x3 = 13. That means one descendant of the 3,4,5 triangle is the 5, 12, 13. You can work out the other two.
If you are paying careful attention, we just added the excenters of the two legs to get the hypotenuse of the triangle... did you know that worked that way? How about this, you also get the hypotenuse if you subtract the top row product from the bottom one. Geometrically that would be subtracting the incenter from the excenter on the hypotenuse... Now go convince yourself that makes perfect sense.
If you find the determinant of a matrix, but ignore the signs and add everything (like we did when we added the diagonals of the 2x2) it is called the Permanent of the matrix. I had never seen this word before. Jeff Miller's web site on the first use of math words provided a little detail. The term seems to have been created by Cauchy. "In his book Permanents  H. Minc mentions that the name permanent is essentially due to Cauchy (1812) although the word as such was first used by Muir in 1882. Nevertheless a referee of one of Minc's earlier papers admonished him for inventing this ludicrous name!"