Roots of the equations `|{:(x,,m,,n,,1),(a,,x,,n,,1),(a,,b,,x,,1),(a,,b,,c,,1):}|=0` are

Options

A. independent of m and n
B. independent of a,b and c
C. depend on m,n and a,b,c
D. inedependent of m,n and a,b,c

Explanation

To solve the equation given by the determinant \( | \begin{pmatrix} x & m & n & 1 \\ a & x & n & 1 \\ a & b & x & 1 \\ a & b & c & 1 \end{pmatrix} | = 0 \), we will follow these steps:

### Step 1: Write the Determinant
We start with the determinant:
\[ D = | \begin{pmatrix} x & m & n & 1 \\ a & x & n & 1 \\ a & b & x & 1 \\ a & b & c & 1 \end{pmatrix} | \]

### Step 2: Apply Row Operations
We will perform row operations to simplify the determinant. We replace:
- \( R_1 \) with \( R_1 - R_2 \)
- \( R_2 \) with \( R_2 - R_3 \)
- \( R_3 \) with \( R_3 - R_4 \)

This gives us the new determinant:
\[ D = | \begin{pmatrix} x - a & m - x & n - n & 0 \\ a - a & x - b & n - x & 0 \\ a - a & b - b & x - c & 0 \\ a - a & b - b & c - 1 & 1 \end{pmatrix} | \]
which simplifies to:
\[ D = | \begin{pmatrix} x - a & m - x & 0 & 0 \\ 0 & x - b & 0 & 0 \\ 0 & 0 & x - c & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} | \]

### Step 3: Expand the Determinant
Now we can expand the determinant along the last column:
\[ D = 1 \cdot | \begin{pmatrix} x - a & m - x \\ 0 & x - b \\ 0 & 0 \\ 0 & 0 \end{pmatrix} | \]
This determinant simplifies to:
\[ D = (x - a)(x - b)(x - c) \]

### Step 4: Set the Determinant to Zero
To find the roots, we set the determinant to zero:
\[ (x - a)(x - b)(x - c) = 0 \]

### Step 5: Solve for Roots
The roots of the equation are:
\[ x = a, \quad x = b, \quad x = c \]

### Conclusion
The roots of the determinant equation are \( x = a, x = b, x = c \).