Wrong result given by matrix multiplication

I was doing a matrix multiplication with 3 matrices: BT * C * B

BT = [-1 0 3; 0 -3 -1; 4 0 -2; 0 -2 4; 1 0 3; 0 3 1; -4 0 2; 0 2 -4]
C = [32 8 0; 8 32 0; 0 0 12]
B = BT'

The result given by Matlab is:

>> BT * C * B

ans =

   140   -12  -200   160    76    12   200  -160
   -12   300   -72   144   -60  -300    72  -144
  -200   -72   560  -160    56    72  -560   160
   160   144  -160   320   128  -144   160  -320
    76   -60    56   128   140    60   -56  -128
    12  -300    72  -144    60   300   -72   144
   200    72  -560   160   -56   -72   560  -160
  -160  -144   160  -320  -128   144  -160   320

However, as I calculated the result by hand it gave me:

   140    60   -56  -128  -140   -60    56   128
    60   300   -72   144   -60  -300    72  -144
   -56   -72   560  -160    56    72  -560   160
  -128   144  -160   320   128  -144   160  -320
  -140   -60    56   128   140    60   -56  -128
   -60  -300    72  -144    60   300   -72   144
    56    72  -560   160   -56   -72   560  -160
   128  -144   160  -320  -128   144  -160   320

As you can see, Matlab gave wrong answers on

ans(2:end, 1)

and

ans(1, 2:end)

but correct on all the other elements ( i.e., ans(1, 1) and ans(2:end, 2:end) ) in the resulting 8-by-8 matrix.

I have checked my hand calculation several times and found nothing wrong in it.

Why do I get this weird behavior? Am I missing something?

Will be very appreciated if someone could correct/clarify me!

Answer 1

[Jan]

The behavior is not weird, but Matlab calculates the correct results. If your manual computations give you another result, the error is on your side. Let's check this. What is your result for BT * C?

BT * C = 
   -32    -8    36
   -24   -96   -12
   128    32   -24
   -16   -64    48
    32     8    36
    24    96    12
  -128   -32    24
    16    64   -48

Now multiply this with B:

  -1   0   4   0   1   0  -4   0
   0  -3   0  -2   0   3   0   2
   3  -1  -2   4   3   1   2  -4   

The element [2,1] is:

   [-24, -96, -12] * [-1; 0; 3] = 24 + 0 - 36 = -12

So your 60 cannot be true.

Or do you start with C * B?

   -32   -24   128   -16    32    24  -128    16
    -8   -96    32   -64     8    96   -32    64
    36   -12   -24    48    36    12    24   -48    

Seriously, do you really think it is possible, that such a severe bug of Matlab has not been detected in the last 25 years? Matlab uses BLAS libraries to do this, which have been optimized by Intel. You can be sure, that they have tested the code exhaustively.

The is a pattern in your calculation:

ans(1, :) == -ans(5, :)
ans(:, 1) == -ans(:, 5)

This pattern does not occur in the real solution.

[hmhuang]

I think it will give the same result either starting from BT * C or C * B to compute BT * C * B.

Did you try to solve say ans(2,1) manually? I always get nothing but 60 rather than -12 given by matlab.

I know it's ridiculous asking a simple mathematics question, but this is indeed where I get stuck...

[hmhuang]

Thank you very much for the answer. I found BT(1,3) was wronly initialized (should be -3 not 3) so as to the corresponding element in matrix B. That's why I got undesired (not wrong) result by matlab only in first row & column. (my fault)

Thank you for your time anyway!

[Jan]

You are welcome. I had the same situation frequently enough to understand, that the computers seem to act illogical.