Basic Arithmetic With Matrices

Due to math content, this page has special requirements (including JavaScript) for full functionality.
With your current viewing scenario, it is not appearing and behaving as it is supposed to!
Please visit Dr. Carol J.V. Fisher's Homepage to learn what this site has to offer.
Watch the "Welcome" video to get started—hope to see you back here soon!

Dr. Carol J.V. Fisher's Homepage

For this exercise, you need ♥ INTERNET EXPLORER 6.0 and above, with MathPlayer installed.♥

BASIC ARITHMETIC WITH MATRICES

Jump right to the exercises!

Before doing this section, you may want to review basic properties of matrices in Introduction to Matrices.

Whenever you get a new mathematical object (like matrices), it's important to develop tools to work with the new object.
In this exercise, you'll learn how to do basic arithmetic operations with matrices:
adding, subtracting, and multiplying by a constant.

Adding and Subtracting Matrices

Matrices can only be added or subtracted when they have the same size.
In this situation, you just add/subtract the corresponding entries.
For example, [- 210 3]  +  [5-46 -7]  =  [-2+5    1+(-4) 0+6 3+(-7)]   =  [3-36 -4]

[- 210 3]  -  [5-46 -7]  =  [-2-5    1-(-4) 0-6 3-(-7)]   =  [-75-6 10]
Precisely, we have:

ADDING AND SUBTRACTING MATRICES

Suppose matrices A and B have the same size.

Then, the sum S=A+B is defined by sij = aij + bij .
The difference D=A-B is defined by
dij = aij - bij .
In particular, matrices with different sizes cannot be added or subtracted.

Multiplying a Matrix by a Constant

It is equally easy to multiply a matrix by a constant; each entry gets multiplied by the constant.
For example, 7 [- 210 3]  =  [7(-2)  7(1) 7(0) 7(3)]   =  [-14 7 0 21]
Precisely, we have:

MULTIPLYING A MATRIX BY A CONSTANT

Let M be a matrix, with members mij .
Let k be a real number.

Then, the new matrix kM has the same size as M ,
and the member in row i and column j is kmij .

Note: The real number multiplier is often called a constant or a scalar.

When working with matrices, it's important to distinguish between the real number 0 and a zero matrix.
To help with this distinction, we will define 0m ×n (zero, with a subscript of m×n )
to mean the zero matrix of size m×n .
You can read 0m ×n aloud as the m×n zero matrix.

Thus, if A is a 2×3 matrix, then A-A= 02 ×3 .

Or, if A is a p×q matrix, then 0A= 0p ×q .

Be aware that many advanced textbooks write simple things like A-A=0 ,
knowing that the audience has enough mathematical maturity to realize that the zero is really the zero matrix with the same size as A .
However, in this exercise, we will be careful to distinguish between the real number zero, and a zero matrix.

EXAMPLE

Let A  =  [2-1 03] , B  =  [-3-5 10] , and C  =  [011 2-10] .
Then,
A+2B  =  [2-1 03]  +  2 [-3-5 10]   =  [2-1 03]  +  [-6-10 20]   =  [-4-11 23]

A+C is not defined.

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.