Determine the product of the two given matrices: $\left[\begin{array}{rr}1 & 0 \\ 2 & -3\end{array}\right]\left[\begin{array}{rr}-3 & 4 \\ 5 & 2\end{array}\right]$

Step 1 ：Given two matrices A and B as follows: \[A = \left[\begin{array}{rr}1 & 0 \\ 2 & -3\end{array}\right]\] and \[B = \left[\begin{array}{rr}-3 & 4 \\ 5 & 2\end{array}\right]\]

Step 2 ：The product of two matrices is calculated by taking the dot product of the rows of the first matrix with the columns of the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Step 3 ：Let's calculate the product of A and B. The first element of the resulting matrix is obtained by multiplying the elements of the first row of A with the elements of the first column of B and adding them together: \(1*(-3) + 0*5 = -3\).

Step 4 ：The second element of the first row is obtained by multiplying the elements of the first row of A with the elements of the second column of B and adding them together: \(1*4 + 0*2 = 4\).

Step 5 ：The first element of the second row is obtained by multiplying the elements of the second row of A with the elements of the first column of B and adding them together: \(2*(-3) + (-3)*5 = -21\).

Step 6 ：The second element of the second row is obtained by multiplying the elements of the second row of A with the elements of the second column of B and adding them together: \(2*4 + (-3)*2 = 2\).

Step 7 ：So, the product of the two given matrices is \[\boxed{\left[\begin{array}{rr}-3 & 4 \\ -21 & 2\end{array}\right]}\]