Find cofactor C(1,1) of the matrix [-2,5,5,3;6,3,8,-2;6,1,6,6;4,6,3,2]

To find the cofactor \( C(1,1) \) of a matrix, we first need to find the minor \( M(1,1) \), which is the determinant of the submatrix obtained by deleting the first row and first column. Then we apply the cofactor formula:

\[
C(1,1) = (-1)^{1+1} \cdot M(1,1) = M(1,1)
\]

### Matrix:
\[
A = \begin{pmatrix}
-2 & 5 & 5 & 3 \\
6 & 3 & 8 & -2 \\
6 & 1 & 6 & 6 \\
4 & 6 & 3 & 2
\end{pmatrix}
\]

### Step 1: Create the submatrix by deleting the first row and first column.

The submatrix is:
\[
\begin{pmatrix}
3 & 8 & -2 \\
1 & 6 & 6 \\
6 & 3 & 2
\end{pmatrix}
\]

### Step 2: Compute the determinant of this 3x3 submatrix.

\[
\text{det} \begin{pmatrix}
3 & 8 & -2 \\
1 & 6 & 6 \\
6 & 3 & 2
\end{pmatrix}
\]

We can compute the determinant using cofactor expansion along the first row:

\[
\text{det} = 3 \cdot \text{det} \begin{pmatrix} 6 & 6 \\ 3 & 2 \end{pmatrix}
- 8 \cdot \text{det} \begin{pmatrix} 1 & 6 \\ 6 & 2 \end{pmatrix}
+ (-2) \cdot \text{det} \begin{pmatrix} 1 & 6 \\ 6 & 3 \end{pmatrix}
\]

Now let's compute each 2x2 determinant:

\[
\text{det} \begin{pmatrix} 6 & 6 \\ 3 & 2 \end{pmatrix} = (6 \cdot 2) - (6 \cdot 3) = 12 - 18 = -6
\]
\[
\text{det} \begin{pmatrix} 1 & 6 \\ 6 & 2 \end{pmatrix} = (1 \cdot 2) - (6 \cdot 6) = 2 - 36 = -34
\]
\[
\text{det} \begin{pmatrix} 1 & 6 \\ 6 & 3 \end{pmatrix} = (1 \cdot 3) - (6 \cdot 6) = 3 - 36 = -33
\]

### Step 3: Substitute the values into the determinant formula:

\[
\text{det} = 3(-6) - 8(-34) + (-2)(-33)
\]
\[
\text{det} = -18 + 272 + 66 = 320
\]

### Step 4: Cofactor \( C(1,1) \)

Since \( C(1,1) = M(1,1) \), the cofactor is:

\[
C(1,1) = 320
\]

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Do you need more details or have any questions?

Here are some related questions you can try:
1. Find the determinant of the entire matrix.
2. Calculate cofactor \( C(1,2) \) for the same matrix.
3. Find the adjugate of the matrix.
4. Compute the inverse of the matrix (if it exists).
5. Determine the rank of the matrix.

**Tip:** When calculating a determinant, cofactor expansion is usually easier along rows or columns with zeros.

Learn how to find the cofactor C(1,1) of the 4x4 matrix [-2,5,5,3;6,3,8,-2;6,1,6,6;4,6,3,2] using the cofactor expansion method. This solution provides a step-by-step guide to calculating minors and determinants. Suitable for students in Grades 10-12.

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