To determine the number of columns and rows after multiplying, look at the following examples:
A=
( 5 6 7 8 )
( 3 4 5 6 )
( 3 6 7 5 )
The order of the matrix (rows first and columns second) = 3 x 4.
Multiply A and another matrix B
A * B
( 5 6 7 8 ) ( 2 3 4)
( 3 4 5 6 ) ( 4 5 2)
( 3 6 7 5 ) ( 4 5 6)
( 0 3 2)
3 x 4 4 x 3
Red represents whether the matrix can be multiplied, blue represents the matrix after multiplication.
Try it out, because it's been a while I haven't touched transformations.