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how do you solve for this involving problem?

The question provided involves several steps related to matrix theory, particularly cofactor matrices, determinants, and inverses. Let’s break down the steps and solve each part.

Given:

Matrix \( E = \begin{pmatrix} 2 & -1 & 2 \\ 1 & -4 & 1 \\ 3 & -1 & 1 \end{pmatrix} \)

The cofactor transpose (also called adjugate matrix) of \( E \) is given by:

\[ F^T = \begin{pmatrix} -3 & -1 & 2 \\ -13 & e & 10 \\ -4 & f & -1 \end{pmatrix} \]

### Part (i): Find the values of \( e \) and \( f \), and hence write the cofactor of matrix \( E \).

1. **Cofactor Matrix Calculation**:
   The cofactor of a matrix is found by calculating the determinant of each 2x2 submatrix, which is formed by removing the corresponding row and column of the element.

   We need to compute cofactors for each element of matrix \( E \). Let’s compute a few key cofactors:

   - **Cofactor for element (1,1):**
     Submatrix by removing the first row and first column of \( E \):
     \[
     \begin{pmatrix} -4 & 1 \\ -1 & 1 \end{pmatrix}
     \]
     Determinant = \( (-4)(1) - (1)(-1) = -4 + 1 = -3 \).

   - **Cofactor for element (2,2):**
     Submatrix by removing the second row and second column of \( E \):
     \[
     \begin{pmatrix} 2 & 2 \\ 3 & 1 \end{pmatrix}
     \]
     Determinant = \( (2)(1) - (2)(3) = 2 - 6 = -4 \).

   - **Cofactor for element (3,3):**
     Submatrix by removing the third row and third column of \( E \):
     \[
     \begin{pmatrix} 2 & -1 \\ 1 & -4 \end{pmatrix}
     \]
     Determinant = \( (2)(-4) - (-1)(1) = -8 + 1 = -7 \).

   To find \( e \) and \( f \), match the elements of the given cofactor transpose matrix with the cofactors you calculated.

2. **Finding \( e \) and \( f \)**:
   From the cofactor matrix and its transpose, you can deduce the values of \( e \) and \( f \) based on the cofactor expansion rules.

### Part (ii): Use the result in (i) to find the determinant of matrix \( E \) by using the elements in the third column.

The determinant of matrix \( E \) can be calculated using cofactor expansion along any column or row. The formula is:

\[
\text{Det}(E) = e_{31}C_{31} + e_{32}C_{32} + e_{33}C_{33}
\]

Here, \( e_{ij} \) represents the element in the matrix, and \( C_{ij} \) represents the cofactor. Perform this calculation along the third column.

### Part (iii): Find the inverse of matrix \( E \) using the adjoint method.

1. **Adjugate Matrix**:
   The adjugate of a matrix is the transpose of its cofactor matrix. Use the cofactor matrix calculated in part (i) to find the adjugate matrix.

2. **Inverse Matrix Formula**:
   The inverse of matrix \( E \) is given by:
   \[
   E^{-1} = \frac{1}{\text{det}(E)} \times \text{adj}(E)
   \]
   Calculate the determinant from part (ii), then divide the adjugate matrix by this determinant to obtain the inverse.

Would you like me to calculate these steps in detail? If yes, I can guide you through each numerical calculation step.

Explore how to solve matrix theory problems involving cofactor matrices, determinants, and matrix inverses step-by-step. Learn the cofactor expansion formula and matrix inverse formula applied to a specific matrix example.

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