The pitfall is your code Q = Transpose[B].M.B // MatrixForm. When you write this way, the MatrixForm gets recorded inside the definition of Q, and it hampers further evaluation. A workaround is a couple of parentheses:

( Q = Transpose[B].M.B) // MatrixForm

Wolfram would make life easier if they finally found a way to make MatrixForm transparent to calculations.

Thank you for your responses. My apologies - I am new here and did try posting the code, but it didn't copy and paste clearly likely because I was trying to show the output as well. I was able to get the matrix multiplication to work, but only after scrapping the entire file and starting over. Now I cannot seem to get a double integration of the resulting matrix to evaluate. The output shows more like a restatement of the problem, and still has variables in the matrix when it should be numerical (they are definite integrals). I am posting the code below, but also attaching a picture so that you can see my results.

Clear [x, y]

Clear [B]

B = ( {
    {(y - 10)/100, 0, (10 - y)/100, 0, y/100, 0, -y/100, 0},
    {0, (x - 10)/100, 0, -x/100, 0, x/100, 0, (10 - x)/100},
    {(x - 10)/100, (y - 10)/100, -x/100, (10 - y)/100, x/100, 
     y/100, (10 - x)/100, -y/100}
   });

Clear [M]

M = ({
    {10666.67, 2666.67, 0},
    {2666.67, 10666.67, 0},
    {0, 0, 4000}
   });

Clear [Q]

Q = Transpose[B].M.B // MatrixForm

Integrate[Q, {x, 0, 10}, {y, 0, 10}]

Attachments:

Please post code rather than a picture (and you're likely to get more and better help). Also, it would be good if you could show what didn't seem to work when you multiplied the three matrices.