# Does a “cubic” matrix exist?

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.

If we’re working with three-dimensional vectors, a matrix is a $3\times 3$ array of 9 numbers. If I’m understanding your question right, you’re asking whether there is something like a $3\times 3\times 3$ 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 $D\times D$ numbers.
• Your “cubic” thing is a “rank-3 tensor”, containing $D\times D\times D$ 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 $f(e_1,e_1)$, $f(e_1,e_2)$, $f(e_1,e_3)$, $f(e_2,e_1)$, 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.