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Завдання 2. Розв’язати систему лінійних рівнянь:
а) за формулами Крамера;
б) методом «оберненої матриці».
Виконати перевірку. 


The given system of linear equations is:

\[
\begin{cases}
3x_1 - x_2 + x_3 = 8, \\
5x_1 + x_2 + 2x_3 = 11, \\
x_1 + 2x_2 + 4x_3 = 4.
\end{cases}
\]

Let's solve this system of equations using both **Cramer's Rule** and the **Inverse Matrix Method**.

### 1. Cramer's Rule
Cramer's Rule allows us to solve a system of linear equations using determinants. For a system of the form \( A\mathbf{x} = \mathbf{b} \), the solution is given by:

\[
x_i = \frac{\text{det}(A_i)}{\text{det}(A)},
\]

where \( A_i \) is the matrix formed by replacing the \(i\)-th column of \( A \) with the vector \( \mathbf{b} \).

### 2. Inverse Matrix Method
The solution to the system \( A\mathbf{x} = \mathbf{b} \) can also be found using the inverse matrix method:

\[
\mathbf{x} = A^{-1} \mathbf{b},
\]

where \( A^{-1} \) is the inverse of the matrix \( A \).

I will now solve the system using both methods.The solutions for the system of equations using both **Cramer's Rule** and the **Inverse Matrix Method** are:

\[
x_1 = 2, \quad x_2 = -1, \quad x_3 = 1.
\]

### Verification
The solutions are the same using both methods, confirming the correctness.

Would you like further details on the steps, or do you have any specific questions?

### Follow-up Questions:
1. What are the general steps to apply Cramer's Rule?
2. How do you find the inverse of a matrix?
3. Can Cramer's Rule be used for larger systems of equations?
4. What are the limitations of the inverse matrix method?
5. How do you check the validity of your solution for a system of equations?

### Tip:
When solving systems of linear equations, always check the determinant of the coefficient matrix first. If it's zero, the system may not have a unique solution, and methods like Cramer's Rule or matrix inversion may not apply.

Learn how to solve a system of linear equations using Cramer's Rule and the Inverse Matrix Method. This problem covers key concepts in linear algebra, including determinants and matrix operations. Suitable for advanced high school students and those studying linear algebra.

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