The given matrix equation is:

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

The task is to find $X$, which is a $2 \times 2$ matrix.

To solve for $X$, we multiply both sides of the equation by the inverse of the matrix $\begin{pmatrix} 3 & 2 \\ -5 & -3 \end{pmatrix}$:

$X = \left(\begin{pmatrix} 3 & 2 \\ -5 & -3 \end{pmatrix}\right)^{-1} \cdot \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}.$

Step 1: Compute the inverse of $\begin{pmatrix} 3 & 2 \\ -5 & -3 \end{pmatrix}$

The formula for the inverse of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is:

$\text{Inverse} = \frac{1}{ad - bc} \cdot \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}.$

For $\begin{pmatrix} 3 & 2 \\ -5 & -3 \end{pmatrix}$, we have:

• $a = 3$, $b = 2$, $c = -5$, $d = -3$,
• Determinant ($ad - bc$) = $(3)(-3) - (2)(-5) = -9 + 10 = 1$.

The inverse is:

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

Step 2: Multiply the inverse with $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

Perform the matrix multiplication:

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

The computation is as follows:

• Top left: $(-3)(1) + (-2)(3) = -3 - 6 = -9$,
• Top right: $(-3)(2) + (-2)(4) = -6 - 8 = -14$,
• Bottom left: $(5)(1) + (3)(3) = 5 + 9 = 14$,
• Bottom right: $(5)(2) + (3)(4) = 10 + 12 = 22$.

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

Final Answer:

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

Let me know if you want further explanation or help!

Related Questions:

1. How is the determinant of a $2 \times 2$ matrix calculated?
2. Why is the inverse of a matrix important in solving linear equations?
3. How do you verify that two matrices are inverses of each other?
4. What happens if the determinant of a matrix is zero?
5. How can you compute the inverse of larger matrices (e.g., $3 \times 3$)?

Tip:

Always double-check the determinant before attempting to find the inverse of a matrix. If the determinant is zero, the matrix does not have an inverse!