If a- b and c are real numbers- and - b-c c-a a-b- c-a a-b b-c- a-b b-c c-a -0-Show that either a-b-c-0 or a-b-

Expanding along R1, we have: Hence, if Δ = 0, then either a + b + c = 0 or a = b = c. ...

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Evaluation of a Determinant

If a, b and c...

Question

If a, b and c are real numbers, and,

Show that either a + b + c = 0 or a = b = c.

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Solution

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a, b

c

Δ = ∣ ∣ ∣ ∣ \begin{matrix} b + c & c + a & a + b c + a & a + b & b + c a + b & b + c & c + a \end{matrix} ∣ ∣ ∣ ∣ = 0

a + b + c = 0

a = b = c

|\begin{matrix} b + c & c + a & a + b \\ c + a & a + b & b + c \\ a + b & b + c & c + a \end{matrix}| = 0

a + b + c = 0 or, a = b = c

a, b, c

a (a - b) + b (b - c) + c (c - a) = 0,

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

A (a, b + c), B (b, c + a), C (c, a + b)

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