The roots of the determinant equation (in x) | vedclass

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Question

(A) Given the determinant equation: $\left| \begin{array}{ccc} a & a & x \ m & m & m \ b & x & b \end{array} ight| = 0$. Expanding the determinant

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$x = a, b$

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(A) Given the determinant equation: $\left| \begin{array}{ccc} a & a & x \ m & m & m \ b & x & b \end{array} ight| = 0$. Expanding the determinant along the second row: $-m \left| \begin{array}{cc} a & x \ x & b \end{array} ight| + m \left| \begin{array}{cc} a & x \ b & b \end{array} ight| - m \left| \begin{array}{cc} a & a \ b & x \end{array} ight| = 0$. Assuming $m 
eq 0$,we can divide by $m$: $-\left( ab - x^2 ight) + \left( ab - bx ight) - \left( ax - ab ight) = 0$. $-ab + x^2 + ab - bx - ax + ab = 0$. $x^2 - (a + b)x + ab = 0$. Factoring the quadratic equation: $(x - a)(x - b) = 0$. Thus,the roots are $x = a$ and $x = b$.

The roots of the determinant equation (in $x$) $\left| \begin{array}{ccc} a & a & x \\ m & m & m \\ b & x & b \end{array} \right| = 0$ are:

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