#color(white)(aaa)[A]color(white)(aaaaaaa)[B]color(white)(aaaaa)[C]#
#[(4,0),(-1,3),(2,-5)] * [(1),(-3)] = [(a), (b),(c)]#
#color(white)(aa)color(red)3" x "color(blue)2color(white)(aaaa)color(blue)2" x "color(red)1color(white)(aa)color(red)3" x "color(red)1#
The size of the first matrix #A# is #"2 x 2"# because it has 2 rows and 2 columns. The size of the second matrix #B# is #"2 x 1"# because it has 2 rows and 1 column.
When the sizes are written next to each other, the "inner" numbers must match. The "outer" numbers determine the size of the resultant matrix.
The resultant matrix #C# has been labeled #[(a),(b),(c)]# and has size #3" x "1#.
Position #a# is in the first row and first column. It's value is obtained by multplying the first row of matrix #A# by the first column of matrix #B#.
#a=4*1 +0*3=4#
Position #b# is in the second row and first column. It is the result of multiplying the second row of #A# by the first column of #B#.
#b=-1*1+3*-3=-10#
Position #c# is obtained by multiplying the third row of #A# by the first column of #B#.
#c=2*1+ -5*-3=17#
The answer is then #[(4),(-10),(17)]#
