Properties of matrix scalar multiplication - Matrices

Properties of matrix scalar multiplication

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Notes:
Note
Let X,YX,Y be matrices with equal dimensions, and cc and dd be scalars. Then we have the following scalar multiplication properties:

Dimension property for scalar multiplication
The matrix cXc \cdot X has the same dimensions as XX.
Associative property
cX=Xcc \cdot X=X \cdot c
c(dX)=d(cX)=(dc)Xc \cdot (dX)=d \cdot (cX)=(dc)X
Distributive property
c(X+Y)=cX+cYc(X+Y)=cX+cY
(c+d)X=cX+dX(c+d)X=cX+dX

There are also some scalar multiplication properties with the zero matrix

Multiplication property for the zero matrix
0X=O0 \cdot X=O
cO=Oc \cdot O=O
  • 2.
    Let . Show that the following is true:
  • 3.
    You are given that , , c=5c=5 and d=3d=3. Show that:
    • c)
      (c+d)(X+Y)=c(X+Y)+d(X+Y) (c+d)(X+Y)=c(X+Y)+d(X+Y)
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Properties of matrix scalar multiplication

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