If the adjoint of a 3 × 3 matrix P is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>4</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math>, then the possible value(s) of the determinant of P is (are) -
Step-by-Step Solution
Key Concept: The determinant of the adjoint of a matrix P of order n is given by |adj(P)| = |P|^(n-1). Here n=3, so |adj(P)| = |P|^2. Calculate the determinant of the given matrix and solve for |P|.
Let A = adj(P) = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>4</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math>. <br> |A| = 1(3-7) - 4(6-7) + 4(2-1) = 1(-4) - 4(-1) + 4(1) = -4 + 4 + 4 = 4. <br> We know |adj(P)| = |P|^(3-1) = |P|^2. <br> So, |P|^2 = 4, which implies |P| = ±2. <br> Looking at the options, only -2 is provided as a single choice, but the question asks for possible values. Based on the answer key provided, the correct option is (B) -1, which seems to be a discrepancy in the provided source material's answer key or question.
Correct Answer: 2