Okay kiddo, do you know what a matrix is? It's like a big box filled with numbers. Now, a persymmetric matrix is a special kind of matrix where if you flip it one way, it stays the same. And if you flip it another way, it looks the same but the numbers are in a different order. It's like turning your drawing upside down and looking at it, everything looks the same but the picture is flipped.
Let's say we have a matrix A which has numbers in it like this:
1 2 3
2 4 5
3 5 6
Now if we take this matrix and flip it left to right or up to down, we get the same matrix. That means it is persymmetric. It looks the same when flipped. This type of matrix is important in math because it has some special properties and makes studying them easier. Hope that helps!