If 3 2 5 4 A é ù-ê ú = ê ú ë û , 1 4 6 7 B é ù ê ú = ê ú-ë û . Find, A – 4B+7I2

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.

Learn how to solve the matrix problem A - 4B + 7I2 by applying matrix subtraction, scalar multiplication, and the identity matrix properties. Suitable for Grades 10-12, this step-by-step solution covers the main concepts and formulas used in matrix operations.

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