The repeated factor of the determinant y+z x yz+x z xx+y y z is

R1→R1+R2+R32(x+y+z)x+y+zx+y+zz+xzxx+yyz=(x+y+z)211z+xzxx+yyzC1→C1−C2−C3=(x+y+z)0110zx(x−z)yz=(x+y+z)(x−z)11zx=(x−z)2⋅(x+y+z)

The repeated factor of the determinant y+z x yz+x z xx+y y z is

R1→R1+R2+R32(x+y+z)x+y+zx+y+zz+xzxx+yyz=(x+y+z)211z+xzxx+yyzC1→C1−C2−C3=(x+y+z)0110zx(x−z)yz=(x+y+z)(x−z)11zx=(x−z)2⋅(x+y+z)

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The repeated factor of the determinant $|\begin{array}{lll} y + z x y \\ z + x z x \\ x + y y z \end{array}|$ is

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$(x - y)$

$(y - z)$

$(x - z)$

$(x + y + z)$

answer is C.

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Detailed Solution

$\begin{array}{l} R_{1} \to R_{1} + R_{2} + R_{3} \\ |\begin{array}{ccc} 2 (x + y + z) & x + y + z & x + y + z \\ z + x & z & x \\ x + y & y & z \end{array}| \\ = (x + y + z) |\begin{array}{ccc} 2 & 1 & 1 \\ z + x & z & x \\ x + y & y & z \end{array}| \end{array}$

$\begin{array}{l} C_{1} \to C_{1} - C_{2} - C_{3} \\ = (x + y + z) \end{array}$

$\begin{array}{l} |\begin{array}{ccc} 0 & 1 & 1 \\ 0 & z & x \\ (x - z) & y & z \end{array}| = (x + y + z) (x - z) |\begin{array}{ll} 1 & 1 \\ z & x \end{array}| \\ = (x - z)^{2} \cdot (x + y + z) \end{array}$