Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|"><mtable><mtr><mtd><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>+</mo><mfrac><mi>a</mi><mi>b</mi></mfrac></mtd><mtd><mfrac><mi>b</mi><mi>c</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mi>c</mi></mfrac></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mfrac><mi>b</mi><mi>c</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mi>c</mi></mfrac></mtd><mtd><mfrac><mi>c</mi><mi>a</mi></mfrac><mo>+</mo><mfrac><mi>c</mi><mi>a</mi></mfrac></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mfrac><mi>c</mi><mi>a</mi></mfrac><mo>+</mo><mfrac><mi>c</mi><mi>a</mi></mfrac></mtd><mtd><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>+</mo><mfrac><mi>a</mi><mi>b</mi></mfrac></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math>, where a, b, c ∈ R+, then which of the following is necessarily true -
(A) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>a</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>b</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>c</mi></mfrac><mo>=</mo><mn>0</mn></math>
(B) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><mo>=</mo><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>c</mi></math>
(C) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>b</mi><mo>=</mo><mi>c</mi></math>
(D) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>a</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>b</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>c</mi></mfrac><mo>=</mo><mn>1</mn></math>
Step-by-Step Solution
Key Concept: The determinant of a matrix with identical rows or columns is zero. If the determinant is zero, it implies that the rows are linearly dependent. For the given determinant, setting a=b=c satisfies the condition, and the expression a^2+b^2+c^2-ab-bc-ca=0 is equivalent to (a-b)^2+(b-c)^2+(c-a)^2=0, which holds when a=b=c.
The given determinant is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|"><mtable><mtr><mtd><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>+</mo><mfrac><mi>a</mi><mi>b</mi></mfrac></mtd><mtd><mfrac><mi>b</mi><mi>c</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mi>c</mi></mfrac></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mfrac><mi>b</mi><mi>c</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mi>c</mi></mfrac></mtd><mtd><mfrac><mi>c</mi><mi>a</mi></mfrac><mo>+</mo><mfrac><mi>c</mi><mi>a</mi></mfrac></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mfrac><mi>c</mi><mi>a</mi></mfrac><mo>+</mo><mfrac><mi>c</mi><mi>a</mi></mfrac></mtd><mtd><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>+</mo><mfrac><mi>a</mi><mi>b</mi></mfrac></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math>. If a=b=c, then each element becomes 2, 2, 1 in each row, making the determinant 0. This implies a=b=c is a solution. Also, a=b=c implies a^2+b^2+c^2 = ab+bc+ca.
Correct Answer: 2, 3