Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub><mi>x</mi></mtd><mtd><msub><mi>a</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub></mtd><mtd><msub><mi>c</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><msub><mi>b</mi><mn>2</mn></msub><mi>x</mi></mtd><mtd><msub><mi>a</mi><mn>2</mn></msub><mi>x</mi><mo>+</mo><msub><mi>b</mi><mn>2</mn></msub></mtd><mtd><msub><mi>c</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mn>3</mn></msub><mo>+</mo><msub><mi>b</mi><mn>3</mn></msub><mi>x</mi></mtd><mtd><msub><mi>a</mi><mn>3</mn></msub><mi>x</mi><mo>+</mo><msub><mi>b</mi><mn>3</mn></msub></mtd><mtd><msub><mi>c</mi><mn>3</mn></msub></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math>, then possible conditions is/are -
(A) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>∀</mo><msub><mi>a</mi><mi>i</mi></msub><mo>,</mo><msub><mi>b</mi><mi>i</mi></msub><mo>,</mo><mtext> where </mtext><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mn>3</mn></math>
(B) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>∀</mo><msub><mi>a</mi><mi>i</mi></msub><mo>,</mo><msub><mi>b</mi><mi>i</mi></msub><mo>,</mo><mtext> where </mtext><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mn>3</mn></math>
(C) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><msub><mi>a</mi><mn>1</mn></msub></mtd><mtd><msub><mi>b</mi><mn>1</mn></msub></mtd><mtd><msub><mi>c</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mn>2</mn></msub></mtd><mtd><msub><mi>b</mi><mn>2</mn></msub></mtd><mtd><msub><mi>c</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mn>3</mn></msub></mtd><mtd><msub><mi>b</mi><mn>3</mn></msub></mtd><mtd><msub><mi>c</mi><mn>3</mn></msub></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math>
(D) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>±</mo><mn>2</mn><mo>∀</mo><msub><mi>a</mi><mi>i</mi></msub><mo>,</mo><msub><mi>b</mi><mi>i</mi></msub><mo>,</mo><mtext> where </mtext><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mn>3</mn></math>
Step-by-Step Solution
Key Concept: Apply column operations C1 -> C1 + C2 to factor out (1+x) or C1 -> C1 - C2 to factor out (1-x). If x=1, C1=C2, determinant is 0. If x=-1, C1=-C2, determinant is 0. If x is not \pm1, the determinant is (1-x^2) times the determinant of the matrix with columns a, b, c.
Applying C1 -> C1 + C2, we get (a_i+b_i)(1+x) in the first column. If x = -1, the first column becomes 0, so the determinant is 0. Applying C1 -> C1 - C2, we get (a_i-b_i)(1-x) in the first column. If x = 1, the first column becomes 0, so the determinant is 0. If the determinant of the matrix with columns a, b, c is 0, the whole expression is 0.
Correct Answer: 1, 2, 3