Multiple choice

Roots of the equation $\begin{vmatrix} x & m & n & 1 \ a & x & n & 1 \ a & b & x & 1 \ a & b & c & 1 \end{vmatrix}=0$ are

  1. independent of $m$ and $n$
  2. independent of $a,b$ and $c$
  3. depend on $m,n$ and $a,b,c$
  4. independent of $m,n$ and $a,b,c$
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A Correct answer
Explanation

Expanding the determinant along the last column or performing row operations shows that the roots are independent of the variables m, n, a, b, and c.

AI explanation

Use row operations to introduce zeros below the main diagonal by replacing R2 with R2 - R4, R3 with R3 - R4, and R4 with R4 - R1. The determinant transforms into the determinant of [[x, m, n, 1], [0, x - b, 0, 0], [0, 0, x - c, 0], [a - x, b - m, c - n, 0]]. Expanding along the fourth column leaves a 3x3 upper triangular block. The remaining determinant evaluates to x(x - b)(x - c). Because the characteristic equation depends entirely on b and c, the roots are completely independent of m and n.

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