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Scalar Multiplication

The matrix obtained by multiplying every element of a matrix A by a scalar λ is called the scalar multiple of A by λ. 
For example: If

Then the product of 3A will be:


Properties of Scalar Multiplication:
All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by scalars.
 
·  If A and B be two matrices of the same order and if k be a scalar, then:
k (A + B) = kA + kB

·  If k1 and k2 are two scalars and if A is a matrix, then:
 (k1 + k2)A = k1A + k2A and k1 (k2A) = k2(k1A)


Example 1: Let , find the product of 4A.


Example 2: Find out whether or not c (Ax) = A (cx) is a valid equation, where c is a scalar, A is a 2 by 2 matrix, and x is a dimension 2 column vector?

Answer: Yes, c (Ax) = A (cx) is a valid equation, because the order of multiplication does not matter for real numbers.

If ,  and the scalar c = c, we have:


and


We can see that c(Ax) and A(cx) both produces the same matrix at the end. Hence c (Ax) = A (cx).
Note: c (Ax) = A (cx) holds true for all matrices that have the correct dimensions so that the product of Ax is valid.

Example 3: Find the product of .

Solution:



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