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The given problem is a matrix equation that needs to be solved. The equation is written as:

\[
X \cdot \begin{pmatrix} 3 & -4 \\ -5 & 6 \end{pmatrix} = \begin{pmatrix} 5 & 1 \\ 2 & 0 \end{pmatrix}
\]

To solve for \(X\), we need to multiply both sides of the equation by the inverse of the matrix on the left. The steps are as follows:

### 1. Find the inverse of matrix \( A = \begin{pmatrix} 3 & -4 \\ -5 & 6 \end{pmatrix} \).
The inverse of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

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

In this case, \( a = 3 \), \( b = -4 \), \( c = -5 \), and \( d = 6 \).

The determinant is:

\[
\text{det}(A) = (3)(6) - (-4)(-5) = 18 - 20 = -2
\]

Thus, the inverse of \( A \) is:

\[
A^{-1} = \frac{1}{-2} \begin{pmatrix} 6 & 4 \\ 5 & 3 \end{pmatrix} = \begin{pmatrix} -3 & -2 \\ -\frac{5}{2} & -\frac{3}{2} \end{pmatrix}
\]

### 2. Multiply both sides of the equation by \( A^{-1} \) on the left:

\[
X = \begin{pmatrix} 5 & 1 \\ 2 & 0 \end{pmatrix} \cdot A^{-1}
\]

Now, perform the matrix multiplication:

\[
X = \begin{pmatrix} 5 & 1 \\ 2 & 0 \end{pmatrix} \cdot \begin{pmatrix} -3 & -2 \\ -\frac{5}{2} & -\frac{3}{2} \end{pmatrix}
\]

### 3. Multiply the matrices:

\[
X = \begin{pmatrix} (5)(-3) + (1)\left(-\frac{5}{2}\right) & (5)(-2) + (1)\left(-\frac{3}{2}\right) \\ (2)(-3) + (0)\left(-\frac{5}{2}\right) & (2)(-2) + (0)\left(-\frac{3}{2}\right) \end{pmatrix}
\]

Simplifying:

\[
X = \begin{pmatrix} -15 - \frac{5}{2} & -10 - \frac{3}{2} \\ -6 + 0 & -4 + 0 \end{pmatrix}
= \begin{pmatrix} -\frac{35}{2} & -\frac{23}{2} \\ -6 & -4 \end{pmatrix}
\]

Thus, the solution is:

\[
X = \begin{pmatrix} -\frac{35}{2} & -\frac{23}{2} \\ -6 & -4 \end{pmatrix}
\]

---

Would you like more details or have any questions on this? 

Here are 5 related questions to consider:
1. How do you find the determinant of a matrix?
2. What are the conditions for a matrix to have an inverse?
3. How do you multiply two matrices together?
4. What is the significance of the determinant being zero?
5. How can matrix equations be used in solving systems of linear equations?

**Tip:** Always check that the determinant of a matrix is non-zero before attempting to find its inverse!

Solve the matrix equation: X * ( 3  -4  -5  6 ) = ( 5  1  2  0 )

Learn how to solve the matrix equation X * ( 3  -4  -5  6 ) = ( 5  1  2  0 ) by finding the inverse of a 2x2 matrix and performing matrix multiplication. This problem covers key concepts in linear algebra, including matrix inverses and the determinant. Suitable for college-level or advanced high school math students.

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