BASIC ARITHMETIC WITH MATRICES

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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.

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