Well, I’ve heard that a “cubic” matrix would exist and I thought: would it be like a magic cube? And more: does it even have a determinant – and other properties? I’m a young student, so… please don’t get mad at me if I’m talking something stupid.

Thank you.

P.S. I’m 14 years old. I don’t know that much about mathematics, but I swear I’ll try to understand your answers. I just know the basics about PreCalculus.

**Answer**

If we’re working with three-dimensional vectors, a matrix is a 3×3 array of 9 numbers. If I’m understanding your question right, you’re asking whether there is something like a 3×3×3 array of 27 numbers with interesting properties.

Yes, there is such a thing; it is called a *tensor*. Tensors are a generalization of both vectors and matrices:

- A number is a “rank-0 tensor”.
- A vector is a “rank-1 tensor”; it contains D numbers when we’re working in D dimensions.
- A matrix is a “rank-2 tensor”, containing D×D numbers.
- Your “cubic” thing is a “rank-3 tensor”, containing D×D×D numbers.

… and so forth.

One use for a rank-3 tensor is if you want to express a function that takes two vectors and produces a third vector, with the property that if you keep any one of the arguments constant, the output is a linear function of the other input. (That is, a *bilinear* mapping from two vectors to one). One familiar example of such a function is the cross product. In order to completely specify such a thing you need 27 numbers, namely the 3 coordinates of each of f(e1,e1), f(e1,e2), f(e1,e3), f(e2,e1), etc. Using linearity to the left and right, this is enough to determine the output for any two input vectors.

I haven’t heard of any generalization of determinants to higher-rank tensors, but I cannot offhand think of a principled reason why one couldn’t exist.

The study of tensors belongs in the field of *multilinear algebra*. It’s quite possible to get at least an undergraduate degree in mathematics without ever hearing about them. If you take physics, you’ll see lots and lots of them, though.

**Attribution***Source : Link , Question Author : Ian Mateus , Answer Author : hmakholm left over Monica*