Matrices & Determinants
Determinants
Grade Class 12
Question:
If x, y, z are distinct digits (0 ≤ x, y, z ≤ 9) & the minimum possible value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>z</mi></mtd><mtd><mn>9</mn><mi>y</mi></mtd><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd><mtd><mi>y</mi></mtd><mtd><mn>9</mn><mi>x</mi></mtd></mtr><mtr><mtd><mn>9</mn><mi>z</mi></mtd><mtd><mi>y</mi></mtd><mtd><mi>x</mi></mtd></mtr></mtable></mfenced></math> is λ, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>λ</mi><mn>83700</mn></mfrac></math> is (where 9x, 9y & 9z are two digits number)
Step-by-Step Solution
Key Concept: Represent the two-digit numbers as 90+x, 90+y, 90+z and evaluate the determinant, then minimize the resulting expression based on the constraints on x, y, z.
The determinant is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>z</mi></mtd><mtd><mn>90</mn><mo>+</mo><mi>y</mi></mtd><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd><mtd><mi>y</mi></mtd><mtd><mn>90</mn><mo>+</mo><mi>x</mi></mtd></mtr><mtr><mtd><mn>90</mn><mo>+</mo><mi>z</mi></mtd><mtd><mi>y</mi></mtd><mtd><mi>x</mi></mtd></mtr></mtable></mfenced></math>. Simplifying this determinant leads to a value that can be minimized by choosing distinct digits x, y, z. The minimum value λ is found to be 753300, so λ/83700 = 9.
Correct Answer: 9