Question

Prove that `|(x + y, y + z, z + x),(z + x, x + y, y + z),(y + z, z + x, x + y)| = 2|(x, y, z),(z, x, y),(y, z, x)|`

Answer 1

L.H.S. = `|(x + y, y + z, z + x),(z + x, x + y, y + z),(y + z, z + x, x + y)|`

Applying R₁ → R₁ + R₂ + R₃, we get

L.H.S. = `|(2(x + y + z), 2(x + y + z),2(x + y + z)),(z + x, x + y, y + z),(y + z, z + x, x + y)|`

Taking 2 common from R₁, we get

L.H.S. = `2|(x + y + z, x + y + z, x + y + z),(z + x, x + y, y + z),(y + z, z + x, x + y)|`

Applying R₁ → R₁ – R₃, we get

L.H.S. = `2|(x, y, z),(z + x, x + y, y + z),(y + z, z + x, x + y)|`

Applying R₂ → R₂ – R₁, we get

L.H.S. = `2|(x, y, z),(z, x, y),(y + z, z + x, x + y)|`

Applying R₃ → R₃ – R₂, we get

L.H.S. = `2|(x, y, z),(z, x, y),(y,  z, x)|`

= R.H.S.