Matrices & Determinants
Determinants
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-<br>(A) a1A1 + b1B1 + c1C1 = Δ<br>(B) a2A2 + b2B2 + c2C2 = Δ<br>(C) a3A3 + b3B3 + c3C3 = Δ<br>(D) a1A2 + b1B2 + c1C2 = Δ
(A) a<sub>1</sub>A<sub>1</sub> + b<sub>1</sub>B<sub>1</sub> + c<sub>1</sub>C<sub>1</sub> = Δ
(B) a<sub>2</sub>A<sub>2</sub> + b<sub>2</sub>B<sub>2</sub> + c<sub>2</sub>C<sub>2</sub> = Δ
(C) a<sub>3</sub>A<sub>3</sub> + b<sub>3</sub>B<sub>3</sub> + c<sub>3</sub>C<sub>3</sub> = Δ
(D) a<sub>1</sub>A<sub>2</sub> + b<sub>1</sub>B<sub>2</sub> + c<sub>1</sub>C<sub>2</sub> = Δ
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, while the sum of the product of elements of any row (or column) with the cofactors of a different 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 (\Delta). However, the sum of the product of elements of a row with the cofactors of a different row is always 0. Therefore, a1A2 + b1B2 + c1C2 = 0, not \Delta. Thus, option (D) is incorrect.
Correct Answer: D