Using the property of determinants and without expanding, prove that:

Question:

Using the property of determinants and without expanding, prove that:

$\left|\begin{array}{lll}x & a & x+a \\ y & b & y+b \\ z & c & z+c\end{array}\right|=0$

Solution:

$\left|\begin{array}{lll}x & a & x+a \\ y & b & y+b \\ z & c & z+c\end{array}\right|=0$

$\left|\begin{array}{lll}x & a & x+a \\ y & b & y+b \\ z & c & z+c\end{array}\right|=\left|\begin{array}{lll}x & a & x \\ y & b & y \\ z & c & z\end{array}\right|+\left|\begin{array}{lll}x & a & a \\ y & b & b \\ z & c & c\end{array}\right|=0+0=0$

[Here, the two columns of the determinants are identical]

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