We are given the following matrices:

$A = \begin{pmatrix} 3 & -2 \\ 5 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 4 \\ 6 & -7 \end{pmatrix}$ We need to calculate $A - 4B + 7I_2$, where $I_2$ is the 2x2 identity matrix:

$I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

Step 1: Compute $4B$

Multiply each element of matrix $B$ by 4:

$4B = 4 \times \begin{pmatrix} 1 & 4 \\ 6 & -7 \end{pmatrix} = \begin{pmatrix} 4 & 16 \\ 24 & -28 \end{pmatrix}$

Step 2: Compute $7I_2$

Multiply each element of the identity matrix $I_2$ by 7:

$7I_2 = 7 \times \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 7 & 0 \\ 0 & 7 \end{pmatrix}$

Step 3: Compute $A - 4B$

Now subtract $4B$ from $A$:

$A - 4B = \begin{pmatrix} 3 & -2 \\ 5 & 4 \end{pmatrix} - \begin{pmatrix} 4 & 16 \\ 24 & -28 \end{pmatrix} = \begin{pmatrix} 3-4 & -2-16 \\ 5-24 & 4-(-28) \end{pmatrix}$

$A - 4B = \begin{pmatrix} -1 & -18 \\ -19 & 32 \end{pmatrix}$

Step 4: Add $7I_2$ to the result

Now add $7I_2$ to the result from Step 3:

$A - 4B + 7I_2 = \begin{pmatrix} -1 & -18 \\ -19 & 32 \end{pmatrix} + \begin{pmatrix} 7 & 0 \\ 0 & 7 \end{pmatrix} = \begin{pmatrix} -1+7 & -18+0 \\ -19+0 & 32+7 \end{pmatrix}$

$A - 4B + 7I_2 = \begin{pmatrix} 6 & -18 \\ -19 & 39 \end{pmatrix}$

Thus, the result of $A - 4B + 7I_2$ is:

$\begin{pmatrix} 6 & -18 \\ -19 & 39 \end{pmatrix}$

Let me know if you need further details or have any questions!

Here are five related questions to expand on this topic:

1. What happens if you change the value of the scalar multiplying matrix $B$?
2. How would this change if we worked with 3x3 matrices instead of 2x2?
3. How is the identity matrix used in matrix algebra beyond this example?
4. What is the significance of scalar multiplication in matrix operations?
5. Can you generalize this process for any matrix $A$, matrix $B$, and scalar $k$?

Tip: When adding or subtracting matrices, always ensure they have the same dimensions!