Matrices & Determinants
Determinants
Grade Class 12
Question:
If 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> and A1, B1, C1 denote the co-factors of a1, b1, c1 respectively, then the value of the determinant <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> is -
(A) Δ
(B) Δ<sup>2</sup>
(C) Δ<sup>3</sup>
(D) 0
Step-by-Step Solution
Key Concept: The determinant of the matrix of cofactors of a matrix A is equal to |A|^(n-1), where n is the order of the matrix.
Let A be the matrix whose determinant is \Delta. The given determinant is the determinant of the adjugate matrix adj(A). We know that |adj(A)| = |A|^(n-1). Here n=3, so |adj(A)| = \Delta^(3-1) = \Delta^2.
Correct Answer: B