# Matrix Closure Property

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Okay my rather terribly written correspondance explains the Closure Property as: "The sum or product of 2 real numbers are real numbers themselves." Now, it is asking me to fill out Multiplication Matrix by Matrix Properties, for the Closure Property the answer is "Only on Square Matrices" -- What the heck? Can someone please elaborate on this?

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Given two nxm matricies, A and B, the only way for you to be able to multiply them together is if n = m (and hence they are square). Next you need to show that A * B results in an nxn matrix as well, which it does by the definition of matrix multiplication.

If you have an n x m, and an m x p matrix, then the result will be an n x p matrix, which would not be in the set of matricies defined as n x m and m x p unless n = m = p, and hence it is not a closed set under matrix multiplication.

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But you can multiple a 3x3 matrix with a 3x1 matrix... You end up with a 3x1 matrix.

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