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The provided matrix equation is:

\[
X \begin{pmatrix} 2 & -1 \\ 5 & -3 \end{pmatrix} = \begin{pmatrix} 11 & -6 \\ 2 & -2 \end{pmatrix}
\]

You are asked to find matrix \( X \) and calculate the sum of its elements.

### Step 1: Solve for matrix \( X \)

To solve for \( X \), multiply both sides of the equation by the inverse of matrix \( \begin{pmatrix} 2 & -1 \\ 5 & -3 \end{pmatrix} \).

Let matrix \( A = \begin{pmatrix} 2 & -1 \\ 5 & -3 \end{pmatrix} \). We need to find \( A^{-1} \), the inverse of \( A \).

The inverse of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

\[
A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

For matrix \( A = \begin{pmatrix} 2 & -1 \\ 5 & -3 \end{pmatrix} \):

- \( a = 2 \), \( b = -1 \), \( c = 5 \), and \( d = -3 \)
- Determinant \( \text{det}(A) = ad - bc = (2)(-3) - (5)(-1) = -6 + 5 = -1 \)

Thus,

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

### Step 2: Multiply both sides by \( A^{-1} \)

Now, multiply the right-hand side matrix \( \begin{pmatrix} 11 & -6 \\ 2 & -2 \end{pmatrix} \) by \( A^{-1} \) to find matrix \( X \):

\[
X = \begin{pmatrix} 11 & -6 \\ 2 & -2 \end{pmatrix} \begin{pmatrix} 3 & -1 \\ 5 & -2 \end{pmatrix}
\]

Let's calculate this multiplication step-by-step:

\[
X = \begin{pmatrix} (11 \cdot 3 + (-6) \cdot 5) & (11 \cdot -1 + (-6) \cdot -2) \\ (2 \cdot 3 + (-2) \cdot 5) & (2 \cdot -1 + (-2) \cdot -2) \end{pmatrix}
\]

Performing the calculations:

- Top left element: \( 11 \cdot 3 + (-6) \cdot 5 = 33 - 30 = 3 \)
- Top right element: \( 11 \cdot -1 + (-6) \cdot -2 = -11 + 12 = 1 \)
- Bottom left element: \( 2 \cdot 3 + (-2) \cdot 5 = 6 - 10 = -4 \)
- Bottom right element: \( 2 \cdot -1 + (-2) \cdot -2 = -2 + 4 = 2 \)

Thus, matrix \( X \) is:

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

### Step 3: Calculate the sum of the elements of \( X \)

The sum of the elements in matrix \( X \) is:

\[
3 + 1 + (-4) + 2 = 2
\]

### Final Answer:

The sum of the elements of matrix \( X \) is \( 2 \).

---

Do you want further details or have any questions about the solution?

Here are five related questions to expand your understanding:
1. How do you find the inverse of a 3x3 matrix?
2. What happens if the determinant of a matrix is zero?
3. Can a non-square matrix have an inverse?
4. How is matrix multiplication different from scalar multiplication?
5. What are some applications of matrix equations in real-world problems?

**Tip:** Always verify the determinant before attempting to find the inverse of a matrix, as a zero determinant means the matrix is singular (non-invertible).

A matrix equation is given: X * [[2, -1], [5, -3]] = [[11, -6], [2, -2]]. Calculate the sum of the elements of matrix X.

Learn how to solve a matrix equation and find the sum of the elements of matrix X. This step-by-step solution covers key matrix concepts, including inverse matrix calculation and matrix multiplication. Suitable for Grades 10-12.

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