The question involves solving for the matrix $X$ in the equation:

$\begin{pmatrix} 3 & 2 \\ -5 & -3 \end{pmatrix} \cdot X = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}.$

Step-by-Step Solution:

1. Rewrite the equation: The equation is: $A \cdot X = B,$ where:

   B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}.$$ To find $$X$$, multiply both sides by $$A^{-1}$$ (the inverse of $$A$$) from the left: $$X = A^{-1} \cdot B.$$

2. Find the inverse of $A$: The inverse of a $2 \times 2$ matrix is calculated as: $A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A),$ where:

   • $\text{det}(A) = (3)(-3) - (2)(-5) = -9 + 10 = 1$,
   • $\text{adj}(A) = \begin{pmatrix} -3 & -2 \\ 5 & 3 \end{pmatrix}$ (swap the diagonal elements and change the sign of the off-diagonal elements).

   Thus:

   = \begin{pmatrix} -3 & -2 \\ 5 & 3 \end{pmatrix}.$$

3. Calculate $X$: Multiply $A^{-1}$ by $B$: $X = A^{-1} \cdot B = \begin{pmatrix} -3 & -2 \\ 5 & 3 \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}.$

   Perform the matrix multiplication:

   X_{12} = (-3)(2) + (-2)(4) = -6 - 8 = -14,$$ $$X_{21} = (5)(1) + (3)(3) = 5 + 9 = 14, \quad X_{22} = (5)(2) + (3)(4) = 10 + 12 = 22.$$ Therefore: $$X = \begin{pmatrix} -9 & -14 \\ 14 & 22 \end{pmatrix}.$$

Final Answer:

$X = \begin{pmatrix} -9 & -14 \\ 14 & 22 \end{pmatrix}.$

Let me know if you want further explanation or a verification step!

Related Questions:

1. How do you calculate the determinant of a matrix?
2. What is the adjugate matrix and how is it used in finding inverses?
3. How do you verify if a matrix is invertible?
4. Can this process be extended to larger matrices, and if so, how?
5. What are the practical applications of solving matrix equations?

Tip: Always check the determinant of a matrix before attempting to find its inverse; a determinant of zero means the matrix is not invertible.