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.
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, [beautiful math coming... please be patient] $$ \begin{bmatrix} 2 & 1\cr 0 & 3 \end{bmatrix} + \begin{bmatrix} 5 & 4\cr 6 & 7 \end{bmatrix} = \begin{bmatrix} 2+5 & 1+(4)\cr 0+6 & 3+(7) \end{bmatrix} = \begin{bmatrix} 3 & 3\cr 6 & 4 \end{bmatrix} $$ [beautiful math coming... please be patient] $$ \begin{bmatrix} 2 & 1\cr 0 & 3 \end{bmatrix}  \begin{bmatrix} 5 & 4\cr 6 & 7 \end{bmatrix} = \begin{bmatrix} 25 & 1(4)\cr 06 & 3(7) \end{bmatrix} = \begin{bmatrix} 7 & 5\cr 6 & 10 \end{bmatrix} $$
Precisely, we have:
WolframAlpha uses braces $\,\{\ \}\,$ to input matrices, not brackets $\,[\ ]\,$.
Then, it displays the matrices (so you can verify your input) using parentheses $\,(\ )\,$.
Get used to this—different academic disciplines and different environments often use different notation.
The screenshot below shows how to input matrices at
wolframalpha.com.
Each row also goes inside a pair of braces $\,\{\ \}\,$, with elements separated by commas, like this: $\,\{2,1\}\,$
The rows themselves are also separated by commas, like this: $\,\{2,1\}\ ,\ \{0,3\}\,$
The entire matrix is enclosed within a pair of braces, like this: $\,\ \{\ \{2,1\}\ ,\ \{0,3\}\ \}\,$
If you want, cutandpaste the following input to WolframAlpha to duplicate what you see in the image below:
{ {2,1} , {0,3} } + { {5,4} , {6,7} }
It is equally easy to multiply a matrix by a constant; each entry gets multiplied by the constant.
For example, [beautiful math coming... please be patient] $$ 7 \begin{bmatrix} 2 & 1\cr 0 & 3 \end{bmatrix} = \begin{bmatrix} 7(2) & 7(1)\cr 7(0) & 7(3) \end{bmatrix} = \begin{bmatrix} 14 & 7\cr 0 & 21 \end{bmatrix} $$
Precisely, we have:
When working with matrices, it's important to distinguish between the real number $\,0\,$ and
a zero matrix.
To help with this distinction, we define
$\,0_{m\times n}\,$
(zero, with a subscript of $\,m\times n\,$)
to mean the zero matrix of size $\,m\times n\,$.
You can read $\,0_{m\times n}\,$ aloud as ‘the $\,m\,$ by $\,n\,$ zero matrix’.
Thus, if $\,A\,$ is a $\,2\times 3\,$ matrix, then: [beautiful math coming... please be patient] $$ AA= 0_{2\times 3} $$ Or, if $\,A\,$ is a $\,p\times q\,$ matrix, then: $$ 0A= 0_{p\times q} $$
Be aware that many advanced textbooks write simple things like
[beautiful math coming... please be patient]
$\,AA=0,\,$
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.
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
