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Scalars & Matrices: Properties & Application

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will review what a matrix is and what a scalar is. After we've done this, we will discuss scalar multiplication and look at an application involving this type of multiplication. We'll finish up by looking at the properties of scalar multiplication.

Matrices

Who doesn't love a sale? Let's suppose that your favorite clothing store XYZ is a store that sells shirts, pants, and shorts. It has two store locations in Locktown and Fernland. The store is quite popular due to its unique fee structure. Every pair of pants is the same price, every shirt is the same price, and every pair of shorts is the same price. The prices vary with location, but each type of clothing has a specific price, regardless of the style.

The prices, based on location, are illustrated in the following table:

Locktown Fernland
Shirts $10 $ 12
Pants $15 $14
Shorts $8 $10

We can actually use a mathematical tool to condense this table into an array of numbers as shown:


scamat1


This rectangular array of numbers is called a matrix in mathematics. This concept has a pretty fancy name, but as we've just seen, it's nothing more than a way to write a collection of numbers in an organized manner.

In our matrix, there are three rows and two columns, so we say that this matrix is a 3 x 2 matrix. In general, an m x n matrix is a rectangular array of numbers with m rows and n columns.


scamat2


That's fairly simple, wouldn't you say?

Scalars and Scalar Multiplication

Now that we know what a matrix is, let's talk about regular numbers when working in matrix land. When we are working in matrix land, we call a number, n, a scalar. We do this so that we don't get a number mixed up with a 1 x 1 matrix. Once again, it's just a fancy name for a very simple concept. A scalar is simply a number.

Alright, back to this clothing store, and didn't we say something about a sale? Suppose we get lucky, and the store decides to hold a storewide sale at both locations, marking everything down by 25%. In other words, everything is 75% of its normal price during the sale. This tells us that if we want to find the sale price of something in the store, we multiply it by 0.75. Since everything in the store is marked down, we need to multiply all the prices by 0.75 to find the sales price of each item.

In terms of our matrix that shows both store locations' prices, we can multiply each entry by the scalar 0.75, and this will give us a matrix displaying all of the sale prices for each item at each location. In mathematics, this is called scalar multiplication, and in general, when we multiply a matrix by a scalar, we simply multiply each entry in the matrix by that scalar.


scamat3


Let's go ahead and apply this to our store's matrix to find those sale prices!


scamat4


We can also put this in tabular form to display the sale prices for each item at each location, if desired.

Locktown Fernland
Shirts $7.50 $9
Pants $11.25 $10.50
Shorts $6 $7.50

Awesome! Now we know exactly how much each item is at each location during this sale! Let's go shopping!

Properties of Scalar Multiplication

Okay, we know that numbers in matrix land are called scalars, and we know that scalar multiplication involves multiplying each entry in a matrix by a scalar. So far, so good! Now, let's look at some different properties that scalar multiplication holds. To describe these properties, let A and B be m x n matrices, and let a and b be scalars.

  1. Associative Property: a(bA) = (ab)A
  2. Commutative Property: aA = Aa
  3. Distributive Property: (a + b)A = aA + bA and a(A + B) = aA + aB
  4. Identity Property: 1A = A
  5. Multiplicative Property of 0: 0A = O, where O is the m x n matrix with all entries equal to 0. O is called the m x n zero matrix.

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