The determinant ( x 2 + 1 ) 2 ( x y + 1 ) 2 ( x z + 1 ) 2 ( x y + 1 ) 2 ( y 2 + 1 ) 2 ( y z + 1 ) 2 ( x z + 1 ) 2 ( y z + 1 ) 2 ( z 2 + 1 ) 2 = k(x-y)^2(y-z)^2(z-x)^2, then k =

The given determinant can be written as the product of two matrices: x 2 + 1 x y + 1 x z + 1 x y + 1 y 2 + 1 y z + 1 x z + 1 y z + 1 z 2 + 1 = x 1 0 y 1 0 z 1 0 × x y z 1 1 1 0 0 0 (This is a simplified representation). By expanding or using properties, the determinant evaluates to 2(x-y)^2(y-z)^2(z-x)^2.

Matrices & Determinants

Properties of determinants

Grade Class 12

Question:

The determinant <br> <math xmlns='http://www.w3.org/1998/Math/MathML'><mfenced open='|' close='|'><mtable><mtr><mtd><msup><mrow><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><msup><mrow><mo>(</mo><mi>x</mi><mi>y</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><msup><mrow><mo>(</mo><mi>x</mi><mi>z</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mi>x</mi><mi>y</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><msup><mrow><mo>(</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><msup><mrow><mo>(</mo><mi>y</mi><mi>z</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mi>x</mi><mi>z</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><msup><mrow><mo>(</mo><mi>y</mi><mi>z</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><msup><mrow><mo>(</mo><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr></mtable></mfenced></math> = k(x-y)^2(y-z)^2(z-x)^2, then k =

(A) 1

(B) 2

(D) 4

Step-by-Step Solution

Key Concept: The determinant can be expressed as the product of two matrices of the form A^T * A, where A is a Vandermonde-like matrix. Specifically, the given determinant is the square of the determinant of a matrix with entries (x_i*x_j + 1).

The given determinant can be written as the product of two matrices: <br> <math xmlns='http://www.w3.org/1998/Math/MathML'><mfenced open='|' close='|'><mtable><mtr><mtd><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mtd><mtd><mi>x</mi><mi>y</mi><mo>+</mo><mn>1</mn></mtd><mtd><mi>x</mi><mi>z</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>x</mi><mi>y</mi><mo>+</mo><mn>1</mn></mtd><mtd><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mtd><mtd><mi>y</mi><mi>z</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>x</mi><mi>z</mi><mo>+</mo><mn>1</mn></mtd><mtd><mi>y</mi><mi>z</mi><mo>+</mo><mn>1</mn></mtd><mtd><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open='|' close='|'><mtable><mtr><mtd><mi>x</mi></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mi>y</mi></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mi>z</mi></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>×</mo><mfenced open='|' close='|'><mtable><mtr><mtd><mi>x</mi></mtd><mtd><mi>y</mi></mtd><mtd><mi>z</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> (This is a simplified representation). By expanding or using properties, the determinant evaluates to 2(x-y)^2(y-z)^2(z-x)^2.

Correct Answer: 2

Question:

Step-by-Step Solution

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