You Dimensional

August 16, 2011 by  
Filed under Die Cutting Machines and Supplies

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How do you go about multiplying three-dimensional matrices, and what are the properties of these matrices?

I have a disagreement with a friend about how to multiply together three-dimensional matrices (essentially cubes of numbers). I say it requires three matrices while she says you only need two. Also, with a 2D matrix you only need to take the outer numbers of the dimensions to find the dimension of the new one. There is no such easy test for 3D matrices. I am interested in these more elaborate forms of math, and so I would like to continue learning about these.
No, I'm sorry, I didn't explain it fully. I meant more of a 3x3x3 matrix where the numbers go up, across, AND at you. I hope this helps.

Suppose you have to multiply two matrices [a]nxm and [b]mxp where n, m, p are positive integers.
The product [a]nxm x [b]mxp is defined but [b]mxp x [a]nxm is not defined.
Suppose [a]nxm x [b]mxp = [c]nxp.

Let a(i,j) denotes the element in the i-th row and j-th column of the matrix [a]nxm.
Let b(i,j) denotes the element in the i-th row and j-th column of the matrix [b]mxp.
Let c(i,j) denotes the element in the i-th row and j-th column of the matrix [c]nxp.

Then,
c(i,j) = a(i,1)*b(1,j) + a(i,2)*b(2,j) +....+ a(i,m)*b(m,j)

This is a general expression for multiplication of any two matrix.

In language this means:
Multiply an element of i-th row of the multiplier matrix with the element (having same row as the column of the element with which it is being multiplied) of the j-th column of the multiplicand matrix and add all such products to get the term in the i-th row and j-th column of the product matrix.

Hope this helps.

If you have further query you are free to email me.

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