Matrices & Determinants
Minors and cofactors
Grade Class 12
Question:
If Δ = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|"><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="|"><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 3x3 matrix A is equal to (det(A))^(n-1), where n is the order of the matrix. Here n=3, so the value is \Delta^(3-1) = \Delta^2.
Let A = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|"><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>. The given determinant is the determinant of the cofactor matrix adj(A)<sup>T</sup>. Since det(adj(A)) = (det(A))<sup>n-1</sup>, and det(adj(A)<sup>T</sup>) = det(adj(A)), the value is \Delta<sup>3-1</sup> = \Delta<sup>2</sup>.
Correct Answer: 2