SOLUTION: I need help with this problem today. Please help me! I believe that you can multiply the following problem, I just don't know the definition of why: Can these matrices be multi

Sorry, I cannot understand your slashes and commas on the second one.

You should not use slashes or commas when typing matrices 
just skip spaces between the elements, and line them up,
and type ᐸpreᐳ at the beginning of your post and type ᐸ/preᐳ at
the end of your post, so the numbers will line up vertically 
since it will be in what is known as "courier new font":

If you meant this:

          [ 3 0] 
  [2 4 0]×[-2 3] 
          [ 1 2]

then yes, they can be multiplied.

If you meant this 

     [2] [ 3 0] 
     [4]×[-2 3] 
     [0] [ 1 2]

then no they cannot.

Rule:  

1.  Find the dimension of the first matrix, by first finding
    the "down" number counting the number of numbers going 
    from top to bottom and then counting the "across" number 
    of numbers going from right to left.  The dimension of the 
    matrix is the "down" number by the "across" number.

2.  If the "across" number of the first matrix equals the "down"
    number of the second matrix, then the two matrices can be 
    multiplied, and the product matrix will have the dimensions
    of the "down" number of the first matrix by the "across" number
    of the second matrix.

If you meant this:

        [ 3 0] 
[2 4 0]×[-2 3] 
        [ 1 2]

Then the first matrix is a "1 down by 3 across" matrix and
the second one is a "3 down by 2 across" matrix.

So yes they can be multiplied because the "across" number of the
first matrix equals the "down" number of the second matrix.  The
product matrix will then be a 1 down by 2 across matrix, that is

 (1 by 3)×(3 by 2) = (1 by 2)

The two inner dimensions are the same (2), so the product matrix
will have the outer two dimensions (1 by 2)

Suppose you have these matrices to multiply:


            [6 3 1 9]
[0 5 1 8 7] [4 7 0 3]
[7 3 4 2 3]×[2 9 9 5]
[7 6 5 6 1] [9 4 2 5]
            [8 3 8 7] 

This is a "3 down by 5 across" times a "5 down by 4 across",

so they can be multiplied because

 (3 by 5)×(5 by 4) = (3 by 4)

the two inner dimensions are the same (5) and the product
matrix will be a 3 by 4 matrix.

However if you reversed them:

[6 3 1 9]
[4 7 0 3] [0 5 1 8 7]         
[4 7 0 3]×[7 3 4 2 3]
[2 9 9 5] [7 6 5 6 1]
[9 4 2 5]
          
then you could not multiply them because 
this is a "5 down by 4 across" times a "3 down by 5 across"

and 

(5 by 4)×(3 by 5)

cannot be multiplied because the two inner dimenisions 4 and 3
are not the same.


If you meant this in the problem you submitted: 

     [2] [ 3 0] 
     [4]×[-2 3] 
     [0] [ 1 2] 

then that would be a "3 down by 1 across" times a "3 down by 2 across"

(3 by 1)×(3 by 2)

and the inner dimensions are not the same, so they cannot
be multiplied.

Hope you understand now.

Edwin