Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
The determinant <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><msup><mi>a</mi><mn>2</mn></msup></mtd><mtd><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mrow><mo>(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mi>b</mi><mi>c</mi></mtd></mtr><mtr><mtd><msup><mi>b</mi><mn>2</mn></msup></mtd><mtd><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mrow><mo>(</mo><mi>c</mi><mo>-</mo><mi>a</mi><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mi>c</mi><mi>a</mi></mtd></mtr><mtr><mtd><msup><mi>c</mi><mn>2</mn></msup></mtd><mtd><msup><mi>c</mi><mn>2</mn></msup><mo>-</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mi>a</mi><mi>b</mi></mtd></mtr></mtable></mfenced></math> is divisible by -
(A) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></math>
(B) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo><mo>(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>)</mo><mo>(</mo><mi>c</mi><mo>+</mo><mi>a</mi><mo>)</mo></math>
(C) <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></math>
(D) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>)</mo><mo>(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>)</mo><mo>(</mo><mi>c</mi><mo>-</mo><mi>a</mi><mo>)</mo></math>
Step-by-Step Solution
Key Concept: Use properties of determinants to simplify the expression by performing row operations like R2 -> R2 - R1 and R3 -> R3 - R1, then factor out common terms.
Let the determinant be \Delta. Performing R2 -> R2 - R1 and R3 -> R3 - R1, we can simplify the second column and factor out terms. The determinant simplifies to a product of factors including (a+b+c), (a-b), (b-c), and (c-a). Thus, it is divisible by (A), (B), and (D).
Correct Answer: 1, 2, 4