https://file.aiquickdraw.com/mathbot/file/1732753519486e04r0w0c.jpg

The given problem involves solving a matrix equation to find matrix \( X \). The equation is:

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

To solve for \( X \), we can rewrite the equation as:

\[
X = \begin{pmatrix} 3 & 2 \\ -5 & -3 \end{pmatrix}^{-1} \cdot \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}
\]

### Steps:
1. **Find the inverse of the matrix** \( A = \begin{pmatrix} 3 & 2 \\ -5 & -3 \end{pmatrix} \):
   The formula for the inverse of a 2x2 matrix is:

   \[
   A^{-1} = \frac{1}{\text{det}(A)} \cdot \begin{pmatrix} d & -b \\ -c & a \end{pmatrix},
   \]

   where \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).

   For \( A = \begin{pmatrix} 3 & 2 \\ -5 & -3 \end{pmatrix} \):
   - Determinant \( \text{det}(A) = (3)(-3) - (-5)(2) = -9 + 10 = 1 \).
   - Inverse:

     \[
     A^{-1} = \begin{pmatrix} -3 & -2 \\ 5 & 3 \end{pmatrix}.
     \]

2. **Multiply the inverse matrix** \( A^{-1} \) with \( B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \):

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

   Performing the matrix multiplication:

   \[
   X = \begin{pmatrix} (-3)(1) + (-2)(3) & (-3)(2) + (-2)(4) \\ (5)(1) + (3)(3) & (5)(2) + (3)(4) \end{pmatrix}
   \]

   \[
   X = \begin{pmatrix} -3 - 6 & -6 - 8 \\ 5 + 9 & 10 + 12 \end{pmatrix}
   \]

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

### Final Answer:
\[
X = \begin{pmatrix} -9 & -14 \\ 14 & 22 \end{pmatrix}.
\]

If you need further explanation or have more questions, let me know! Here are related questions for deeper understanding:

1. How is the determinant of a 2x2 matrix calculated?
2. Why do we use the inverse of a matrix to solve such equations?
3. Can this method be extended to higher-order matrices?
4. What happens if the determinant of a matrix is zero?
5. How can we verify the solution of a matrix equation?

**Tip:** Always check the

Solve the matrix equation \( A \cdot X = B \), where \( A = \begin{pmatrix} 3 & 2 \\ -5 & -3 \end{pmatrix} \) and \( B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \).

Learn how to solve a matrix equation \( A \cdot X = B \) where \( A = \begin{pmatrix} 3 & 2 \\ -5 & -3 \end{pmatrix} \) and \( B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \). This solution demonstrates how to find the inverse of a matrix and perform matrix multiplication. Suitable for high school students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation