Let the matrices be
[ begin{bmatrix} a & b \ c & d end{bmatrix} ]
[ begin{bmatrix} e & f \ g & h end{bmatrix} ]
Multiply row 1 of the first matrix by column 1 of the second matrix: (ae + bg)
Multiply row 1 of the first matrix by column 2 of the second matrix: (af + bh)
Multiply row 2 of the first matrix by column 1 of the second matrix: (ce + dg)
Multiply row 2 of the first matrix by column 2 of the second matrix: (cf + dh)
The product is
[ begin{bmatrix} ae + bg & af + bh \ ce + dg & cf + dh end{bmatrix} ]