We are given two matrices $A$ and $B$, and we need to find the expression $A - 4B + 7I_2$, where $I_2$ is the $2 \times 2$ identity matrix.

Step 1: Define the given matrices

$A = \begin{pmatrix} 3 & 2 \\ 5 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 4 \\ -6 & 7 \end{pmatrix}$ The identity matrix $I_2$ is: $I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

Step 2: Calculate $4B$

Multiply matrix $B$ by 4:

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

Step 3: Calculate $7I_2$

Multiply the identity matrix $I_2$ by 7:

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

Step 4: Compute $A - 4B + 7I_2$

Now, subtract $4B$ from $A$, and then add $7I_2$:

$A - 4B = \begin{pmatrix} 3 & 2 \\ 5 & 4 \end{pmatrix} - \begin{pmatrix} 4 & 16 \\ -24 & 28 \end{pmatrix} = \begin{pmatrix} 3 - 4 & 2 - 16 \\ 5 - (-24) & 4 - 28 \end{pmatrix} = \begin{pmatrix} -1 & -14 \\ 29 & -24 \end{pmatrix}$

Now, add $7I_2$ to the result:

$A - 4B + 7I_2 = \begin{pmatrix} -1 & -14 \\ 29 & -24 \end{pmatrix} + \begin{pmatrix} 7 & 0 \\ 0 & 7 \end{pmatrix} = \begin{pmatrix} -1 + 7 & -14 + 0 \\ 29 + 0 & -24 + 7 \end{pmatrix} = \begin{pmatrix} 6 & -14 \\ 29 & -17 \end{pmatrix}$

Final Answer:

$A - 4B + 7I_2 = \begin{pmatrix} 6 & -14 \\ 29 & -17 \end{pmatrix}$

Would you like further details or have any questions?

Here are 5 questions to expand this topic:

1. How do you compute the product of two matrices?
2. What are the properties of the identity matrix?
3. How do you determine if two matrices are equal?
4. Can the determinant of a matrix help in solving systems of equations?
5. What is the significance of matrix transformations in geometry?

Tip: Remember that the identity matrix acts like 1 in matrix operations—multiplying any matrix by the identity matrix leaves the original matrix unchanged.