From analyzing the content of the image, here’s a detailed explanation of the mathematical process shown:

The main focus here appears to be solving a system of linear equations using matrix analysis—specifically, the concept of determinants to determine whether the system is compatible (has solutions) or incompatible (no solutions).

Steps and Explanation:

1. Matrix $A$ and Its Determinant:
   • The matrix $A$ is given as: \begin{bmatrix} 2 & -3 & 1 \\ 1 & 2 & -1 \\ 2 & 2 & 1 \end{bmatrix}.$$
   • The determinant of $A$ ($\text{det}(A)$) is calculated using the formula for a 3x3 matrix: a(ei − fh) − b(di − fg) + c(dh − eg).$$ Substituting the elements of $$A$$: $$\text{det}(A) = 2 \cdot (2 \cdot 1 - (-1) \cdot 2) - (-3) \cdot (1 \cdot 1 - (-1) \cdot 2) + 1 \cdot (1 \cdot 2 - 2 \cdot 2).$$ Simplifying: $$\text{det}(A) = 2 \cdot (2 + 2) - (-3) \cdot (1 + 2) + 1 \cdot (2 - 4),$$ $$\text{det}(A) = 2 \cdot 4 + 3 \cdot 3 - 2,$$ $$\text{det}(A) = 8 + 9 - 2 = 15.$$ So, $$\text{det}(A) = 15$$.
   • Since $\text{det}(A) \neq 0$, matrix $A$ is invertible, and the system of linear equations is compatible (it has a unique solution).

2. Setting Up the System:
   • The augmented matrix for the system appears to be given, but details of the specific equations and right-hand side vector ($\mathbf{b}$) aren't fully clear in this image. Nonetheless, the determinant's non-zero value ensures that the system has a unique solution, and no further checks for compatibility are needed.

3. System Analysis (Alternative Scenario):

   • The image also shows a scenario where the determinant is zero. When $\text{det}(A) = 0$, it is concluded that the system is incompatible (no solution) or possibly dependent (infinitely many solutions). This depends on the rank of the augmented matrix.

   • However, as calculated, $\text{det}(A) \neq 0$, so the system remains compatible with a unique solution.

4. Conclusion:
   • The determinant $\text{det}(A)$ serves as the deciding factor for the solvability of the system. Here, because $\text{det}(A) \neq 0$, the matrix is invertible, and the system is consistent with a unique solution.

Additional Concepts Highlighted:

• Determinant Properties:

  • A determinant of 0 indicates the matrix is singular (non-invertible), leading to an incompatible or dependent system.
  • A non-zero determinant implies an invertible matrix, guaranteeing a unique solution.

• System Compatibility:

  • Compatible systems have at least one solution (unique or infinite).
  • Incompatible systems have no solution.

Do you want me to clarify any specific part of this explanation, or go further into solving a similar system?

Related Questions:

1. How is the determinant of a 3x3 matrix calculated step by step?
2. What are the implications of $\text{det}(A) = 0$ for the solution of a linear system?
3. What methods other than determinants can be used to analyze a system's compatibility (e.g., row reduction)?
4. How does the rank of a matrix relate to the solution of a system of equations?
5. What are the geometrical interpretations of compatible and incompatible systems?

Tip:

When working with determinants and matrices, always verify your computations step by step to avoid errors in sign or arithmetic, as these can change the outcome significantly.