Question

Without expanding the determinants, show that `|(x"a", y"b", z"c"),("a"^2, "b"^2, "c"^2),(1, 1, 1)| = |(x, y, z),("a", "b", "c"),("bc", "ca", "ab")|`

Answer 1

L.H.S. = `|(x"a", y"b", z"c"),("a"^2, "b"^2, "c"^2),(1, 1, 1)|`

= Taking a, b, c common from C₁, C₂, C₃ respectively, we get

L.H.S. = `"abc"|(x, y, z),("a", "b", "c"),(1/"a", 1/"b", 1/"c")|`

= `|(x, y, z),("a", "b", "c"),("abc"/"a", "abc"/"b", "abc"/"c")|`

= `|(x, y, z),("a", "b", "c"),("bc", "ca", "ab")|`

= R.H.S.