Find the values of the following determinants
`|{:(1,1,1),(5,-3,1),(7,4,-2):}|`

Answer

To find the value of the determinant

\[ \Delta = \begin{vmatrix} 1 & 1 & 1 \\ 5 & -3 & 1 \\ 7 & 4 & -2 \end{vmatrix} \]

we will follow these steps:

### Step 1: Write down the determinant
We start with the determinant as given:

\[ \Delta = \begin{vmatrix} 1 & 1 & 1 \\ 5 & -3 & 1 \\ 7 & 4 & -2 \end{vmatrix} \]

### Step 2: Apply column transformations
We will perform column operations to simplify the determinant. Specifically, we will replace the second column with the second column minus the first column, and the third column with the third column minus the first column.

1. **Column 2**: \(C_2 \leftarrow C_2 - C_1\)
2. **Column 3**: \(C_3 \leftarrow C_3 - C_1\)

After performing these operations, we get:

\[ \Delta = \begin{vmatrix} 1 & 0 & 0 \\ 5 & -8 & -4 \\ 7 & -3 & -9 \end{vmatrix} \]

### Step 3: Expand the determinant
Now we can expand the determinant along the first row. The determinant can be calculated as follows:

\[ \Delta = 1 \cdot \begin{vmatrix} -8 & -4 \\ -3 & -9 \end{vmatrix} \]

### Step 4: Calculate the 2x2 determinant
Now we will calculate the 2x2 determinant:

\[ \begin{vmatrix} -8 & -4 \\ -3 & -9 \end{vmatrix} = (-8)(-9) - (-4)(-3) = 72 - 12 = 60 \]

### Step 5: Combine results
Thus, we have:

\[ \Delta = 1 \cdot 60 = 60 \]

### Final Result
The value of the determinant is:

\[ \Delta = 60 \]

---