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**Unformatted text preview: **2 = 3 · 2. We know from algebra4 that 3x = x + x + x, so it
seems natural that given a matrix A, we deﬁne 3A = A + A + A. If A = [aij ]m×n , we have
3A = A + A + A = [aij ]m×n + [aij ]m×n + [aij ]m×n = [aij + aij + aij ]m×n = [3aij ]m×n
In other words, multiplying the matrix in this fashion by 3 is the same as multiplying each entry
by 3. This leads us to the following deﬁnition.
Definition 8.8. Scalara Multiplication: We deﬁne the product of a real number and a matrix
to be the matrix obtained by multiplying each of its entries by said real number. More speciﬁcally,
if k is a real number and A = [aij ]m×n , we deﬁne
kA = k [aij ]m×n = [kaij ]m×n
a
The word ‘scalar’ here refers to real numbers. ‘Scalar multiplication’ in this context means we are multiplying
a matrix by a real number (a scalar). One may well wonder why the word ‘scalar’ is used for ‘real number.’ It has everything to do with
‘scaling’ factors.5 A point P (x, y ) in the plane can be represented by its position matrix, P :
(x, y ) ↔ P = x
y Suppose we take the point (−2, 1) and multiply its position matrix by 3. We have...

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