Multiple choice

The roots of the determinantal equation $\begin{vmatrix} a &a &x \ m& m & m\ b& x &b \end{vmatrix}$ =0 are

  1. $a$
  2. $b$
  3. $a$ or $b$
  4. $0,0$
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C Correct answer
Explanation

Expanding the determinant: a(mb-mx) - a(mb-mb) + x(mx-mb) = a(mb-mx) + x(mx-mb) = amb - amx + xmx - xmb = mb(a-x) - mx(a-x) = (a-x)(mb-mx) = m(a-x)(b-x). Setting to 0 gives x = a or x = b.

AI explanation

Perform row operations on the determinant by subtracting the second row from the third row. The determinant becomes the determinant of [[a, a, x], [m, m, m], [b - m, x - m, b - m]]. Factor out m from the second row. Expanding the determinant along this second row yields -m times the determinant of [[b - m, b - m], [b - m, x - m]] plus m times the determinant of [[a, x], [b - m, b - m]]. Simplifying this gives m(a - b)(x - b) = 0. Since m cannot be assumed to be zero, the roots of the equation are x = a and x = b.

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