We are tasked with multiplying the matrices $A$ and $B$, where:

$A = \begin{pmatrix} 2 & -3 \\ 1 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 2 \\ 1 & -1 \end{pmatrix}.$

The matrix multiplication $A \times B$ is calculated as:

$C = A \cdot B = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix},$

where:

$a_{11} = (2)(1) + (-3)(1), \quad a_{12} = (2)(2) + (-3)(-1),$ $a_{21} = (1)(1) + (4)(1), \quad a_{22} = (1)(2) + (4)(-1).$

Step-by-step Calculation:

1. $a_{11} = 2 \cdot 1 + (-3) \cdot 1 = 2 - 3 = -1$,
2. $a_{12} = 2 \cdot 2 + (-3) \cdot (-1) = 4 + 3 = 7$,
3. $a_{21} = 1 \cdot 1 + 4 \cdot 1 = 1 + 4 = 5$,
4. $a_{22} = 1 \cdot 2 + 4 \cdot (-1) = 2 - 4 = -2$.

Resulting Matrix:

$C = A \cdot B = \begin{pmatrix} -1 & 7 \\ 5 & -2 \end{pmatrix}.$

Let me know if you would like further clarifications or additional examples.

Expanding Questions:

1. How do you determine the dimensions of a resulting matrix in matrix multiplication?
2. What are some common applications of matrix multiplication in real-world problems?
3. Can the order of matrices in multiplication be reversed? Why or why not?
4. How would the result differ if the elements in $A$ or $B$ were transposed?
5. What are the conditions for two matrices to be conformable for multiplication?

Tip: Always ensure the number of columns in the first matrix matches the number of