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Find the value of x if

Question:

Find the value of x if

$\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2\end{array}\right]\left[\begin{array}{l}1 \\ 2 \\ x\end{array}\right]=\mathrm{O}$

Solution:

Given, $\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2\end{array}\right]\left[\begin{array}{l}1 \\ 2 \\ x\end{array}\right]=O$

$\Rightarrow\left[\begin{array}{lll}1+2 x+15 & 3+5 x+3 & 2+x+2\end{array}\right]\left[\begin{array}{l}1 \\ 2 \\ x\end{array}\right]=O$

$\Rightarrow[16+2 x 5 x+6 x+4]\left[\begin{array}{l}1 \\ 2 \\ x\end{array}\right]=O$

[16 + 2x + 10x + 12 + x2 + 4x] = 0

 

[x2 + 16x + 28] = 0

 

x2 + 16x + 28 = 0

(x + 2) (x + 14) = 0

Therefore, x = -2, -14

$A=\left[\begin{array}{cc}5 & 3 \\ -1 & -2\end{array}\right]$

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