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The problem asks to solve the matrix equation:

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

To solve for \( X \), we need to multiply both sides of the equation by the inverse of the matrix \( A = \begin{pmatrix} 3 & -4 \\ -5 & 6 \end{pmatrix} \), assuming it is invertible. 

### Step 1: Find the inverse of matrix \( A \)

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

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

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

The determinant \( \text{det}(A) = ad - bc = (3)(6) - (-4)(-5) = 18 - 20 = -2 \).

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

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

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

Now multiply both sides of the equation by \( A^{-1} \) from the right:

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

### Step 3: Perform matrix multiplication

We now multiply the matrices:

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

For element \( (1,1) \):
\[
X_{11} = (5)(-3) + (1)(-\frac{5}{2}) = -15 - \frac{5}{2} = -\frac{30}{2} - \frac{5}{2} = -\frac{35}{2}
\]

For element \( (1,2) \):
\[
X_{12} = (5)(-2) + (1)(-\frac{3}{2}) = -10 - \frac{3}{2} = -\frac{20}{2} - \frac{3}{2} = -\frac{23}{2}
\]

For element \( (2,1) \):
\[
X_{21} = (2)(-3) + (0)(-\frac{5}{2}) = -6
\]

For element \( (2,2) \):
\[
X_{22} = (2)(-2) + (0)(-\frac{3}{2}) = -4
\]

Thus, the matrix \( X \) is:

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

This is the solution to the matrix equation.

---

Would you like more details on any of the steps or have further questions? Here are some related questions:

1. How do you compute the determinant of larger matrices?
2. How can we verify the solution by multiplying back the matrices?
3. What happens if the determinant of a matrix is zero?
4. Can you explain matrix multiplication in more detail?
5. How is the inverse matrix method applied to systems of linear equations?

**Tip:** Always verify your answer by multiplying the solution matrix back with the original one!

Solve the matrix equation: X * [[3, -4], [-5, 6]] = [[5, 1], [2, 0]]

Solve a matrix equation involving matrix inversion and multiplication: X * [[3, -4], [-5, 6]] = [[5, 1], [2, 0]]. This problem explores key linear algebra concepts, including finding the inverse of a matrix and performing matrix multiplication. Suitable for advanced high school students.

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