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(A) Given that $A$ is a singular matrix,its determinant must be zero,i.e.,$|A| = 0$. $|A| = \begin{vmatrix} x & 1 & 2 \ 2 & 4 & x \ -3 & 3 & 2 \end{

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-$9$

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(A) Given that $A$ is a singular matrix,its determinant must be zero,i.e.,$|A| = 0$. $|A| = \begin{vmatrix} x & 1 & 2 \ 2 & 4 & x \ -3 & 3 & 2 \end{vmatrix} = 0$ Expanding along the first row: $x(4 	imes 2 - 3 	imes x) - 1(2 	imes 2 - (-3) 	imes x) + 2(2 	imes 3 - (-3) 	imes 4) = 0$ $x(8 - 3x) - 1(4 + 3x) + 2(6 + 12) = 0$ $8x - 3x^2 - 4 - 3x + 36 = 0$ $-3x^2 + 5x + 32 = 0$ Multiplying by $-1$: $3x^2 - 5x - 32 = 0$ For the quadratic equation $ax^2 + bx + c = 0$,the sum of roots $x_1 + x_2 = -b/a$ and the product of roots $x_1 x_2 = c/a$. Here,$x_1 + x_2 = -(-5)/3 = 5/3$ and $x_1 x_2 = -32/3$. Therefore,$x_1 + x_2 + x_1 x_2 = 5/3 - 32/3 = -27/3 = -9$.

If $A = \begin{bmatrix} x & 1 & 2 \\ 2 & 4 & x \\ -3 & 3 & 2 \end{bmatrix}$ is a singular matrix and the distinct values of $x$ are $x_1$ and $x_2$,then $x_1 + x_2 + x_1 x_2 = $.

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