Matrices & Determinants
Minors and cofactors
Grade Class 12
Question:
If in the determinant Δ = <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></math>, A1, B1, C1 etc. be the co-factors of a1, b1, c1 etc., then which of the following relations is incorrect-
(A) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>1</mn></msub><msub><mi>A</mi><mn>1</mn></msub><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub><msub><mi>B</mi><mn>1</mn></msub><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><msub><mi>C</mi><mn>1</mn></msub><mo>=</mo><mo>∆</mo></math>
(B) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>2</mn></msub><msub><mi>A</mi><mn>2</mn></msub><mo>+</mo><msub><mi>b</mi><mn>2</mn></msub><msub><mi>B</mi><mn>2</mn></msub><mo>+</mo><msub><mi>c</mi><mn>2</mn></msub><msub><mi>C</mi><mn>2</mn></msub><mo>=</mo><mo>∆</mo></math>
(C) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>3</mn></msub><msub><mi>A</mi><mn>3</mn></msub><mo>+</mo><msub><mi>b</mi><mn>3</mn></msub><msub><mi>B</mi><mn>3</mn></msub><mo>+</mo><msub><mi>c</mi><mn>3</mn></msub><msub><mi>C</mi><mn>3</mn></msub><mo>=</mo><mo>∆</mo></math>
(D) <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub><msub><mi>B</mi><mn>2</mn></msub><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><msub><mi>C</mi><mn>2</mn></msub><mo>=</mo><mo>∆</mo></math>
Step-by-Step Solution
Key Concept: The sum of the product of elements of any row (or column) with their corresponding cofactors is equal to the value of the determinant (\Delta). The sum of the product of elements of any row (or column) with the cofactors of any other row (or column) is zero.
The property of determinants states that the sum of the product of elements of a row with their corresponding cofactors is equal to the determinant value. If we multiply elements of one row with cofactors of another row, the sum is zero. Thus, a1A2 + b1B2 + c1C2 = 0, not \Delta. Therefore, option (D) is incorrect.
Correct Answer: 4