Given two matrices $A=\left[\begin{array}{cc}2& 3\\ 1& 4\end{array}\right]$ and $B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$, find the product $AB$.

Step 1 ：Step 1: Let's compute the element at the first row and the first column of $AB$. The element is the dot product of the first row of $A$ and the first column of $B$, which is $2\ast 5+3\ast 7=10+21=31$.

Step 2 ：Step 2: Compute the element at the first row and the second column of $AB$. The element is the dot product of the first row of $A$ and the second column of $B$, which is $2\ast 6+3\ast 8=12+24=36$.

Step 3 ：Step 3: Compute the element at the second row and the first column of $AB$. The element is the dot product of the second row of $A$ and the first column of $B$, which is $1\ast 5+4\ast 7=5+28=33$.

Step 4 ：Step 4: Compute the element at the second row and the second column of $AB$. The element is the dot product of the second row of $A$ and the second column of $B$, which is $1\ast 6+4\ast 8=6+32=38$.

Step 5 ：Step 5: The product $AB$ is then the matrix formed by these four elements: $\left[\begin{array}{cc}31& 36\\ 33& 38\end{array}\right]$.