Solution By Steps

Step 1: Identify the given matrices

Given matrices are

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} and

B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}.

Step 2: Calculate the product of matrices A and B

To find

AB, we multiply each element of the rows of

A by the corresponding elements of the columns of

B and sum them up.

AB = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} (1 \cdot 5 + 2 \cdot 7) & (1 \cdot 6 + 2 \cdot 8) \\ (3 \cdot 5 + 4 \cdot 7) & (3 \cdot 6 + 4 \cdot 8) \end{pmatrix}

Step 3: Compute the elements of the product matrix

AB = \begin{pmatrix} (5 + 14) & (6 + 16) \\ (15 + 28) & (18 + 32) \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}

Final Answer

The product of matrices

A and

B is

\boxed{\begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}}.