Prove-that-determinant-1-1-1-a-b-c-a-2-b-2-c-2-a-b-b-c-c-a-

Question Number 175110 by nadovic last updated on 19/Aug/22

Prove that | \begin{matrix} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{matrix} | = (a - b) (b - c) (c - a)

Commented by kaivan.ahmadi last updated on 19/Aug/22

| \begin{matrix} 1 1 1 \\ a b c \\ a^{2} b^{2} c^{2} \end{matrix} | | \begin{matrix} 1 1 1 \\ a b c \\ a^{2} b^{2} c^{2} \end{matrix} | =

({b c}^{2} + {c a}^{2} + {a b}^{2}) - ({b a}^{2} + {a c}^{2} + {c b}^{2})

= c (b c + a^{2} - a c - b^{2}) - a b (a - b)

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

= (a - b) (- c^{2} + a c + b c - a b)

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

= (a - b) ((c - a) (b - c)) =

(a - b) (b - c) (c - a)

n o t e t h a t t h i s i s a V a n d e r m o n d e

d e t e r m i n a n t .

I n g e n e r a l c a s e

| \begin{matrix} 1 1 \dots 1 \\ a_{1} a_{2} \dots a_{n} \\ ⋮ ⋮ ⋮ \\ a_{1}^{n - 1} \dots a_{n}^{n - 1} \end{matrix} | = \prod_{1 ⩽ i < j ⩽ n} (a_{j} - a_{i})

Commented by Tawa11 last updated on 19/Aug/22

Great sir .

Commented by puissant last updated on 21/Aug/22

V a n d e r m o n d e m e t h o d

Answered by Rasheed.Sindhi last updated on 19/Aug/22

Prove that | \begin{matrix} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{matrix} | = (a - b) (b - c) (c - a)

\overset{―}{\underset{―}{∣} \underset{―}{\frac{}{} Through properties \frac{}{} ∣}}

= | \begin{matrix} 1 & 0 & 0 \\ a & b - a & c - a \\ a^{2} & b^{2} - a^{2} & c^{2} - a^{2} \end{matrix} | \underset{C 3 - C 1}{C 2 - C 1}

= (b - a) (c - a) | \begin{matrix} 1 & 0 & 0 \\ a & 1 & 1 \\ a^{2} & b + a & c + a \end{matrix} |

= (b - a) (c - a) \cdot 1 {1 (c + a) - 1 (b + a)}

= (b - a) (c - a) (c - b)

= (a - b) (b - c) (c - a)

Commented by puissant last updated on 21/Aug/22

G r e a t

Commented by Rasheed.Sindhi last updated on 21/Aug/22

T h a n k s s i r!

Answered by Rajpurohith last updated on 10/Jun/23

A p p l y t h e o p e r a t i o n C_{1} \to C_{1} - C_{2} a n d C_{2} \to C_{2} - C_{3}

T h e d e t e r m i n a n t r e d u c e s t o,

| \begin{matrix} 0 0 1 \\ a - b b - c c \\ a^{2} - b^{2} b^{2} - c^{2} c^{2} \end{matrix} | = (a - b) (b - c) | \begin{matrix} 0 0 1 \\ 1 1 c \\ a + b b + c c^{2} \end{matrix} |

T h i s i s b y t a k i n g t h e c o m m o n f a c t o r s f r o m C_{1} & C_{2}

a n d o p e r a t e C_{1} \to C_{1} - C_{2}

= (a - b) (b - c) | \begin{matrix} 0 0 1 \\ 0 1 c \\ a - c b + c c^{2} \end{matrix} |

e x p a n d i n g a l o n g t h e f i r s t r o w,

= (a - b) (b - c) (c - a)