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$A=\left(\begin{matrix}x & 5 & x \\ 1 & 3 & -2\\ -2 &-2 &2 \end{matrix}\right)$

$B=\left(\begin{matrix}0 & 0 & 21 \\ 1 & -1 & -14\\ 0 &\frac{4}{3} &4\end{matrix}\right)$

Given if B can be obtained From A by applying finitely many basic row operation then what is value of x ?

One thing I did not understand the value of x be unique ![]

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If this is true then we have $\det{(A)}=\det{(B)}$ hence $$6x-2x+20+6x-4x-10=28$$ $$6x=18$$ $$x=3$$

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The point here must be that the "basic row operations" defined in the exercise are not the same as the usual elementary row operations. Namely, multiplying a row by a constant is not allowed.

This means that "basic row operations" cannot change the determinant of the matrix.

Can you find an $x$ such that the two matrices have the same determinant?

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