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To evaluate the determinant of the given 4x4 matrix:

\[
\begin{vmatrix}
2 & -1 & 6 & 4 \\
7 & 2 & -2 & 5 \\
4 & -2 & 10 & 8 \\
6 & 1 & 1 & 4
\end{vmatrix}
\]

We can expand the determinant along the first row. The formula for the determinant of a 4x4 matrix is:

\[
\text{Determinant} = a_{11} \cdot C_{11} - a_{12} \cdot C_{12} + a_{13} \cdot C_{13} - a_{14} \cdot C_{14}
\]

Where \( C_{ij} \) is the cofactor of the element \( a_{ij} \).

### Step 1: Calculate Cofactor \( C_{11} \)

Remove the first row and the first column:

\[
C_{11} = \begin{vmatrix}
2 & -2 & 5 \\
-2 & 10 & 8 \\
1 & 1 & 4
\end{vmatrix}
\]

Expanding this determinant:

\[
C_{11} = 2 \begin{vmatrix} 10 & 8 \\ 1 & 4 \end{vmatrix} - (-2) \begin{vmatrix} -2 & 8 \\ 1 & 4 \end{vmatrix} + 5 \begin{vmatrix} -2 & 10 \\ 1 & 1 \end{vmatrix}
\]

Now, calculate each 2x2 determinant:

\[
\begin{vmatrix} 10 & 8 \\ 1 & 4 \end{vmatrix} = 10(4) - 8(1) = 40 - 8 = 32
\]

\[
\begin{vmatrix} -2 & 8 \\ 1 & 4 \end{vmatrix} = (-2)(4) - (8)(1) = -8 - 8 = -16
\]

\[
\begin{vmatrix} -2 & 10 \\ 1 & 1 \end{vmatrix} = (-2)(1) - (10)(1) = -2 - 10 = -12
\]

Thus:

\[
C_{11} = 2(32) + 2(-16) + 5(-12) = 64 - 32 - 60 = -28
\]

### Step 2: Calculate Cofactor \( C_{12} \)

Remove the first row and the second column:

\[
C_{12} = \begin{vmatrix}
7 & -2 & 5 \\
4 & 10 & 8 \\
6 & 1 & 4
\end{vmatrix}
\]

Expanding this determinant:

\[
C_{12} = 7 \begin{vmatrix} 10 & 8 \\ 1 & 4 \end{vmatrix} + 2 \begin{vmatrix} 4 & 8 \\ 6 & 4 \end{vmatrix} + 5 \begin{vmatrix} 4 & 10 \\ 6 & 1 \end{vmatrix}
\]

Calculate each 2x2 determinant:

\[
\begin{vmatrix} 10 & 8 \\ 1 & 4 \end{vmatrix} = 32
\]

\[
\begin{vmatrix} 4 & 8 \\ 6 & 4 \end{vmatrix} = 4(4) - 8(6) = 16 - 48 = -32
\]

\[
\begin{vmatrix} 4 & 10 \\ 6 & 1 \end{vmatrix} = 4(1) - 10(6) = 4 - 60 = -56
\]

Thus:

\[
C_{12} = 7(32) - 2(-32) + 5(-56) = 224 + 64 - 280 = 8
\]

### Step 3: Calculate Cofactor \( C_{13} \)

Remove the first row and the third column:

\[
C_{13} = \begin{vmatrix}
7 & 2 & 5 \\
4 & -2 & 8 \\
6 & 1 & 4
\end{vmatrix}
\]

Expanding this determinant:

\[
C_{13} = 7 \begin{vmatrix} -2 & 8 \\ 1 & 4 \end{vmatrix} - 2 \begin{vmatrix} 4 & 8 \\ 6 & 4 \end{vmatrix} + 5 \begin{vmatrix} 4 & -2 \\ 6 & 1 \end{vmatrix}
\]

Calculate each 2x2 determinant:

\[
\begin{vmatrix} -2 & 8 \\ 1 & 4 \end{vmatrix} = -16
\]

\[
\begin{vmatrix} 4 & 8 \\ 6 & 4 \end{vmatrix} = -32
\]

\[
\begin{vmatrix} 4 & -2 \\ 6 & 1 \end{vmatrix} = 14
\]

Thus:

\[
C_{13} = 7(-16) - 2(-32) + 5(14) = -112 + 64 + 70 = 22
\]

### Step 4: Calculate Cofactor \( C_{14} \)

Remove the first row and the fourth column:

\[
C_{14} = \begin{vmatrix}
7 & 2 & -2 \\
4 & -2 & 10 \\
6 & 1 & 1
\end{vmatrix}
\]

Expanding this determinant:

\[
C_{14} = 7 \begin{vmatrix} -2 & 10 \\ 1 & 1 \end{vmatrix} - 2 \begin{vmatrix} 4 & 10 \\ 6 & 1 \end{vmatrix} + (-2) \begin{vmatrix} 4 & -2 \\ 6 & 1 \end{vmatrix}
\]

Calculate each 2x2 determinant:

\[
\begin{vmatrix} -2 & 10 \\ 1 & 1 \end{vmatrix} = -12
\]

\[
\begin{vmatrix} 4 & 10 \\ 6 & 1 \end{vmatrix} = -56
\]

\[
\begin{vmatrix} 4 & -2 \\ 6 & 1 \end{vmatrix} = 14
\]

Thus:

\[
C_{14} = 7(-12) - 2(-56) - 2(14) = -84 + 112 - 28 = 0
\]

### Step 5: Compute the Determinant

Finally, the determinant is:

\[
\text{Determinant} = 2(-28) - (-1)(8) + 6(22) - 4(0) = -56 + 8 + 132 = 84
\]

So, the determinant of the matrix is **84**.

Would you like more details or have any questions?

### Related Questions:
1. How do you calculate the determinant of a 3x3 matrix?
2. What is the significance of the determinant in linear algebra?
3. How can you use row reduction to simplify the calculation of a determinant?
4. What does it mean if the determinant of a matrix is zero?
5. How can you find the inverse of a matrix using its determinant?

### Tip:
When computing determinants of large matrices, it's helpful to choose the row or column with the most zeros to simplify the calculations.

Learn how to evaluate the determinant of a 4x4 matrix with detailed steps. This problem covers matrix algebra concepts, including the formula for the determinant of a 4x4 matrix and properties of determinants. Suitable for students in Grades 10-12.

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