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# If the following matrix is singular, what is the value of $x: \; \begin{bmatrix} -3& -5 &1 \\ 9 & 14 & 1\\ 7+x&29 & -2 \end{bmatrix}$

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• For a matrix $A$ to be singular, the value of its determinant, $det (A) = 0$
Given the singular matrix $A = \begin{bmatrix} -3& -5 &1 \\ 9 & 14 & 1\\ 7+x&29 & -2 \end{bmatrix} \rightarrow det (A) = 0$
$\Rightarrow det (A) = -3 \times (14 \times -2 - 29 \times 1) + (-5) \times (7 \times 1 +x \times 1- 9 \times -2) +1 \times (9 \times 29 - 7 \times 14 - x\times 14) =0$
$\Rightarrow det (A) = 171 - 125 + 163 - 5x - 14x = 209 - 19x$
$\Rightarrow det (A) = 0 \rightarrow 209 - 19x = 0 \rightarrow x = 11$

edited Dec 22, 2013