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Question
Evaluate: `|(3x, -x + y, -x + z),(x - y, 3y, z - y),(x - z, y - z, 3z)|`
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Solution
We have, `|(3x, -x + y, -x + z),(x - y, 3y, z - y),(x - z, y - z, 3z)|`
[Applying C1 → C1 + C2 + C3]
= `|(x + y + z, -x + y, -x + z),(x + y + z, 3y, z - y),(x + y + z, y - z, 3z)|`
[Taking (x + y + z) common from colmn C1]
= `(x + y + z)|(1, -x + y, -x + z),(1, 3y, z - y),(1, y - z, 3z)|`
[Applying R1 → R2 – R1 and R3 → R3 – R1]
= `(x + y + z)|(1, -x + y, -x + z),(0, 2y + x, x - y),(0, x - z, 2z + x)|`
[Applying C2 → C2 – C3]
= `(x + y + z)|(1, -x + y, -x + z),(0, 3y, x - y),(0, -3z, 2z + x)|`
[Expanding along first column]
= `(x + y + z) * 1[3y(2z + x) + (3z)(x - y)]`
= (x + y + z)(3yz + 3yx + 3xz)
= 3(x + y + z)(xy + yz + zx)
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