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The image contains a system of linear equations involving variables \( i_1 \), \( i_2 \), and \( i_3 \), as well as an explanation of solving the system using matrices. Here’s an outline of what’s present in the image:

### Given System of Equations:
The goal is to find \( i_1 \), \( i_2 \), and \( i_3 \).

From **(4)**:
1. \( 2(i_3 - i_2) + 5(i_3 - i_1) = 24 \)
2. \( (i_2 - i_3) + 2i_2 + (i_2 - i_1) = 0 \)
3. \( 5(i_1 - i_3) + 2(i_1 - i_2) + i_1 = 6 \)

From **(5)**:
1. \( 2i_1 + 3i_2 + 8i_3 = 30 \)
2. \( 6i_1 - i_2 + 2i_3 = 7 \)
3. \( 3i_1 - 12i_2 + 8i_3 = 0 \)

### Method for Solving the System:
The instructions explain how to solve this system using matrices:
- Represent the system of equations in matrix form as \( A\mathbf{i} = B \), where \( A \) is the coefficient matrix and \( \mathbf{i} \) is the vector of unknowns \( i_1, i_2, i_3 \).
- The solution is given by:
  \[
  \mathbf{i} = A^{-1} \cdot B
  \]
  where \( A^{-1} \) is the inverse of matrix \( A \).
  
The matrix inverse is computed using the **adjoint** (adjugate) method:
- \( A^{-1} = \frac{\text{Adj}(A)}{\det(A)} \)
- The adjoint of \( A \) is \( \text{Adj}(A) = C_A^T \), where \( C_A \) is the cofactor matrix of \( A \) and \( C_A^T \) is its transpose.

Would you like me to proceed with solving this system of equations using this matrix method? Let me know if you'd prefer a breakdown of the steps.

### Here are 5 questions to expand your understanding:
1. What is the significance of finding the inverse of a matrix in solving a system of linear equations?
2. How is the cofactor matrix \( C_A \) computed for a given matrix?
3. What conditions must be met for the matrix \( A \) to be invertible?
4. Could you simplify the system of equations before converting it into matrix form?
5. How can we verify that the matrix solution satisfies the original equations?

#### Tip:
To efficiently solve systems of equations, always check whether the matrix is singular (non-invertible) by calculating its determinant. If the determinant is zero, alternative methods like Gaussian elimination must be used.

Solve the system of equations involving variables i1, i2, and i3 using matrix methods.

This page explains how to solve a system of linear equations using the matrix inversion method. Learn about matrix adjugate, cofactor matrix, and the determinant, applied to solve the variables i1, i2, and i3. Suitable for students in Grades 10-12.

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