We are tasked with finding the value(s) of $m$ such that the matrix

$X = \begin{bmatrix} -3 & -4 \\ -1 & m \end{bmatrix}$

is a linear combination of the matrices in the set $M$ provided:

$M = \left\{ \begin{bmatrix} 1 & -2 \\ 1 & 1 \end{bmatrix}, \begin{bmatrix} 2 & 1 \\ 1 & -2 \end{bmatrix}, \begin{bmatrix} 1 & -7 \\ 2 & 5 \end{bmatrix} \right\}.$

This means there exist scalars $\alpha, \beta, \gamma$ such that:

$X = \alpha \begin{bmatrix} 1 & -2 \\ 1 & 1 \end{bmatrix} + \beta \begin{bmatrix} 2 & 1 \\ 1 & -2 \end{bmatrix} + \gamma \begin{bmatrix} 1 & -7 \\ 2 & 5 \end{bmatrix}.$

Thus, we need to solve the following system of equations for $\alpha, \beta, \gamma,$ and $m$:

$\begin{bmatrix} -3 & -4 \\ -1 & m \end{bmatrix} = \alpha \begin{bmatrix} 1 & -2 \\ 1 & 1 \end{bmatrix} + \beta \begin{bmatrix} 2 & 1 \\ 1 & -2 \end{bmatrix} + \gamma \begin{bmatrix} 1 & -7 \\ 2 & 5 \end{bmatrix}.$

Expanding both sides, we get:

$\begin{bmatrix} -3 & -4 \\ -1 & m \end{bmatrix} = \begin{bmatrix} \alpha + 2\beta + \gamma & -2\alpha + \beta - 7\gamma \\ \alpha + \beta + 2\gamma & \alpha - 2\beta + 5\gamma \end{bmatrix}.$

Now, equating the corresponding elements from both sides:

1. $\alpha + 2\beta + \gamma = -3$
2. $-2\alpha + \beta - 7\gamma = -4$
3. $\alpha + \beta + 2\gamma = -1$
4. $\alpha - 2\beta + 5\gamma = m$

We will now solve this system of equations to find $m$.

Step 1: Solve the system

From the first three equations, solve for $\alpha, \beta, \gamma$, then substitute into the fourth equation to find $m$.

1. $\alpha + 2\beta + \gamma = -3$
2. $-2\alpha + \beta - 7\gamma = -4$
3. $\alpha + \beta + 2\gamma = -1$

Let me solve this step by step.The solutions for $\alpha$ and $\beta$ in terms of $\gamma$ are:

$\alpha = 1 - 3\gamma, \quad \beta = \gamma - 2.$

Now, substitute these expressions into the fourth equation:

$\alpha - 2\beta + 5\gamma = m,$

to find $m$. Let's proceed with this substitution.After solving, we find that $m = 5$.

Thus, the value of $m$ that makes the matrix $X$ a linear combination of the given matrices in $M$ is $\boxed{5}$.

Would you like further details or have any questions?

Here are five related questions for further practice:

1. What is a linear combination in the context of vector spaces?
2. How can the method of solving systems of equations be generalized?
3. What happens if there are more matrices in the set $M$?
4. How can we check if a matrix belongs to a given subspace?
5. What is the geometric interpretation of solving for $m$?

Tip: When dealing with linear combinations, always set up a system of equations by comparing corresponding matrix entries and solve for the unknowns step by step.