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).