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Let's solve the problem step by step.

### Given:
- Matricea \( A = \begin{pmatrix} 7 & 3 \\ 3 & 1 \end{pmatrix} \)
- Matricea \( B = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \)
- Matricea \( I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)

### Problem Statements:

**a) Arătați că \(\det A = -2\).**

**b) Arătați că \( A - 4I_2 = 3B \).**

---

### a) Calculating \(\det A\)

The determinant of matrix \( A \) is calculated using the formula for a 2x2 matrix:

\[
\det A = ad - bc
\]

Where for matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \):

\[
\det A = (7 \times 1) - (3 \times 3) = 7 - 9 = -2
\]

Thus, \(\det A = -2\), which satisfies the condition.

---

### b) Demonstrating that \( A - 4I_2 = 3B \)

First, let's calculate \( 4I_2 \) and then find \( A - 4I_2 \):

\[
4I_2 = 4 \times \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix}
\]

Now subtract \( 4I_2 \) from \( A \):

\[
A - 4I_2 = \begin{pmatrix} 7 & 3 \\ 3 & 1 \end{pmatrix} - \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix} = \begin{pmatrix} 7-4 & 3-0 \\ 3-0 & 1-4 \end{pmatrix} = \begin{pmatrix} 3 & 3 \\ 3 & -3 \end{pmatrix}
\]

Next, let's check if \( A - 4I_2 = 3B \):

\[
3B = 3 \times \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} = \begin{pmatrix} 3 & 3 \\ 3 & -3 \end{pmatrix}
\]

As you can see:

\[
A - 4I_2 = \begin{pmatrix} 3 & 3 \\ 3 & -3 \end{pmatrix} = 3B
\]

Therefore, the condition \( A - 4I_2 = 3B \) is satisfied.

---

### Summary:
- The determinant of matrix \( A \) is indeed \(-2\).
- The equation \( A - 4I_2 = 3B \) holds true.

Would you like more details on any of the steps? 

Here are five related questions you might want to explore:
1. What is the determinant of the matrix \( B \)?
2. How do you find the inverse of matrix \( A \)?
3. Can the equation \( A - 4I_2 = 3B \) be generalized for other matrices?
4. How do eigenvalues relate to the determinant of a matrix?
5. What are the properties of matrix multiplication that were used in this problem?

**Tip:** When working with matrix equations, always check if the operations are valid by ensuring the matrices are of compatible dimensions.

Learn how to solve matrix equations using the example A - 4I = 3B. Understand matrix algebra concepts including determinants and matrix operations. Suitable for advanced high school students and those interested in matrix mathematics.

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