We are given the following matrices in the problem:

$B = \begin{bmatrix} 1 & -4 \\ 3 & 5 \end{bmatrix}, \quad C = \begin{bmatrix} 2 & -5 \\ 1 & 7 \end{bmatrix}$ and it is asked to find a matrix $X$, where $X - C = 2B$.

Step 1: Rearrange the equation

We start with the given equation: $X - C = 2B$ To isolate $X$, add $C$ to both sides: $X = 2B + C$

Step 2: Compute $2B$

Now, let's compute $2B$: $2B = 2 \times \begin{bmatrix} 1 & -4 \\ 3 & 5 \end{bmatrix} = \begin{bmatrix} 2 & -8 \\ 6 & 10 \end{bmatrix}$

Step 3: Add $2B$ and $C$

Next, add the matrices $2B$ and $C$: $X = \begin{bmatrix} 2 & -8 \\ 6 & 10 \end{bmatrix} + \begin{bmatrix} 2 & -5 \\ 1 & 7 \end{bmatrix}$ $X = \begin{bmatrix} 2+2 & -8+(-5) \\ 6+1 & 10+7 \end{bmatrix} = \begin{bmatrix} 4 & -13 \\ 7 & 17 \end{bmatrix}$

Thus, the matrix $X$ is: $X = \begin{bmatrix} 4 & -13 \\ 7 & 17 \end{bmatrix}$

Let me know if you need further details or explanations.

Related Questions:

1. What are the properties of matrix addition and scalar multiplication used here?
2. How do we multiply a matrix by a scalar?
3. What is the determinant of matrix $X$ and how is it calculated?
4. Can we find the inverse of matrix $X$, and if so, how?
5. How do we verify that matrix $X$ satisfies the equation $X - C = 2B$?

Tip:

When adding or subtracting matrices, always ensure the dimensions are compatible (both matrices must have the same number of rows and columns).