the multiplication of 2 matrices A and B
a11 a12 a13 ...
a21 a22 a23 ...
a31 a32 a33 ...
...
b11 b12 b13 ...
b21 b22 b23 ...
b31 b32 b33 ...
is
a11×b11+a12×b21+a13×b31+... a11×b12+a12×b22+a13×b32 ...
a21×b11+a22×b21+a23×b31+... a21×b12+a22×b22+a23×b32
a31×b11+a32×b21+a33×b31+... a31×b12+a32×b22+a33×b32
...
in other words, we combine the rows of the first matrix with the columns of the second matrix.
the rows of the result matrix are driven by the rows of the first matrix and the columns of the second matrix.
formally :
we can multiply a m×n matrix only with a n×o matrix.
the number of columns in the first matrix must be equal to the number of rows in the second matrix.
the result matrix is then sized m×o.
the first element (first row first column) of the result matrix is the sum of the products of the elements of the first row of the first matrix with the elements of the first column of the second matrix.
the element in the (x,y) position in the result matrix is the sum of the products of the elements of the xth row of the first matrix with the elements of the yth column of the second matrix.
so, when multiplying 2 2×2 matrices, the condition is automatically fulfilled : m = n = o = 2
ST =
4×0+1×-4 4×4+1×3
-3×0+-2×-4 -3×4+-2×3
=
-4 19
8 -18
and TS =
0×4+4×-3 0×1+4×-2
-4×4+3×-3 -4×1+3×-2
=
-12 -8
-25 -10
as we can see, the 2 results are completely different.