Solve the following determinant equations: x+1&3&5 2&x+2&5 2&3&x+… - Vidyadip

Question

Solve the following determinant equations:
$\begin{vmatrix}\text{x}+1&3&5\\2&\text{x}+2&5\\2&3&\text{x}+4\end{vmatrix}=0$

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Answer

Let $\begin{vmatrix}\text{x}+1&3&5\\2&\text{x}+2&5\\2&3&\text{x}+4\end{vmatrix}$
$=\begin{vmatrix}\text{x}+9&3&5\\\text{x}+9&\text{x}+2&5\\\text{x}+9&3&\text{x}+4\end{vmatrix}$ [Applying $C_1 = C_1 + C_2 + C_3]$
$=(\text{x}+9)\begin{vmatrix}1&3&5\\1&\text{x}+2&5\\1&3&\text{x}+4\end{vmatrix}$
$=(\text{x}+9)\begin{vmatrix}1&3&5\\0&\text{x}-1&0\\1&3&\text{x}+4\end{vmatrix}=0$ [Applying $R_2 \rightarrow R_2 - R_1]$
$=(\text{x}+9)\begin{vmatrix}1&3&5\\0&\text{x}-1&0\\1&0&\text{x}-1\end{vmatrix}$ [Applying $R_3 \rightarrow R_3 - R_1]$
$=(\text{x}+9)(\text{x}-1)^2=0$
$\text{x}=-9,1,1$

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