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Scalar Multiplication of Matrices

Reference > Mathematics > Algebra > Matrices
 

In this section, you will learn a new vocabulary word. The word isĀ scalar. If you look up definitions of scalar, you might see scary things like this:

  • A non-vector quantity
  • A quantity having magnitude, but not direction
  • The real component of a quarternion
  • The reduction of a vector, matrix or tensor to a single component

If these definitions scare you, don't worry; I have a simpler definition to give you:

  • A scalar is a number.

There. Wasn't that easy?

So multiplication of a scalar and a matrix simply means, multiplying a matrix by a number. Would you like to guess how we do that? It's pretty easy and straightforward; you just multiply each entry of the matrix by the scalar.

Example One
If A =
1
4
5
and k = 4, calculate kA.

Solution
Mutliply each element of A times k: kA =
4
16
20

Example Two
Calculate 6
1
3
1
2
2
2
3

Solution
Multiply each element of the matrix by 6:
2
3
12
4

Example Three
Find x and y if x
1
5
6
=
6
y
36

Solution
We can easily see that x = 6, since 1x = 6 and 6x = 36 (the first and third column products).

Looking at the second column, 5x = y, so 5(6) = y = 30.

Questions

1.
Explain the difference between a scalar and a matrix
2.
Find k if k
5
7
=
35
49
3.
Find the value of x if x
x-1
0
=
0
0
4.
If the product of a scalar and a matrix is a matrix with zero entries throughout, what can you conclude?
5.
If k = 3, h = 2, A =
5
2
, B =
3
4
, what is the first column, first column entry of kA + hB?
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Adding Matrices and Subtracting MatricesAdding Matrices and Subtracting Matrices
Matrix MultiplicationMatrix Multiplication
 

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