Question

If the matrix $A=\left[\begin{array}{ccc}1& 3& x+2\\ 2& 4& 8\\ 3& 5& 10\end{array}\right]$ is singular, then x = _______________.

Answer 1

Given: The matrix $A=\left[\begin{array}{ccc}1& 3& x+2\\ 2& 4& 8\\ 3& 5& 10\end{array}\right]$ is singular

$$\begin{gathered} A\mathrm{is}\mathrm{singular}\Rightarrow \left|A\right|=0 \\ \mathrm{Thus}, \\ \left|\begin{array}{ccc}1& 3& \mathrm{x}+2\\ 2& 4& 8\\ 3& 5& 10\end{array}\right|=0 \\ \Rightarrow 1\left(40-40\right)-3\left(20-24\right)+\left(\mathrm{x}+2\right)\left(10-12\right)=0 \\ \Rightarrow 1\left(0\right)-3\left(-4\right)+\left(\mathrm{x}+2\right)\left(-2\right)=0 \\ \Rightarrow 12-2\mathrm{x}-4=0 \\ \Rightarrow 8-2\mathrm{x}=0 \\ \Rightarrow 2\mathrm{x}=8 \\ \Rightarrow \mathrm{x}=4 \end{gathered}$$

Hence, x = 4.