Let x 1 , x 2 , x 3 and y 1 , y 2 , y 3 be distinct real numbers. The value of the determinant D = ( x 1 − y 1 ) 2 ( x 2 − y 1 ) 2 ( x 3 − y 1 ) 2 ( x 1 − y 2 ) 2 ( x 2

Question

Let x 1 ​ , x 2 ​ , x 3 ​ and y 1 ​ , y 2 ​ , y 3 ​ be distinct real numbers. The value of the determinant D = ​ ( x 1 ​ − y 1 ​ ) 2 ( x 2 ​ − y 1 ​ ) 2 ( x 3 ​ − y 1 ​ ) 2 ​ ( x 1 ​ − y 2 ​ ) 2 ( x 2 ​ − y 2 ​ ) 2 ( x 3 ​ − y 2 ​ ) 2 ​ ( x 1 ​ − y 3 ​ ) 2 ( x 2 ​ − y 3 ​ ) 2 ( x 3 ​ − y 3 ​ ) 2 ​ ​ is:

TestHub · Accepted Answer

Each element of the determinant is of the form $(x_{i} - y_{j})^{2} = x_{i}^{2} - 2 x_{i} y_{j} + y_{j}^{2}$ . This determinant can be expressed as a product of two matrices. Let $A$ be the given determinant. $A = x_{1}^{2} x_{2}^{2} x_{3}^{2} - 2 x_{1} - 2 x_{2} - 2 x_{3} 111 111 y_{1} y_{2} y_{3} y_{1}^{2} y_{2}^{2} y_{3}^{2}$

Let $M_{1} = x_{1}^{2} x_{2}^{2} x_{3}^{2} - 2 x_{1} - 2 x_{2} - 2 x_{3} 111$ and $M_{2} = 111 y_{1} y_{2} y_{3} y_{1}^{2} y_{2}^{2} y_{3}^{2}$ . Then $D = ∣ M_{1} ∣ \cdot ∣ M_{2} ∣$ . $∣ M_{1} ∣ = (- 2) x_{1}^{2} x_{2}^{2} x_{3}^{2} x_{1} x_{2} x_{3} 111 = (- 2) (x_{1} - x_{2}) (x_{2} - x_{3}) (x_{3} - x_{1})$ . $∣ M_{2} ∣ = 111 y_{1} y_{2} y_{3} y_{1}^{2} y_{2}^{2} y_{3}^{2} = (y_{2} - y_{1}) (y_{3} - y_{1}) (y_{3} - y_{2})$ . Thus, $D = (- 2) (x_{1} - x_{2}) (x_{2} - x_{3}) (x_{3} - x_{1}) (y_{2} - y_{1}) (y_{3} - y_{1}) (y_{3} - y_{2})$ . This can be rewritten as $2 (x_{1} - x_{2}) (x_{2} - x_{3}) (x_{3} - x_{1}) (y_{1} - y_{2}) (y_{2} - y_{3}) (y_{3} - y_{1})$ .

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