Let a determinant is given by A = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mi>p</mi></mtd><mtd><mi>q</mi></mtd><mtd><mi>r</mi></mtd></mtr><mtr><mtd><mi>x</mi></mtd><mtd><mi>y</mi></mtd><mtd><mi>z</mi></mtd></mtr></mtable></mfenced></math> and suppose A = 6. If B = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>p</mi><mo>+</mo><mi>x</mi></mtd><mtd><mi>q</mi><mo>+</mo><mi>y</mi></mtd><mtd><mi>r</mi><mo>+</mo><mi>z</mi></mtd></mtr><mtr><mtd><mi>a</mi><mo>+</mo><mi>x</mi></mtd><mtd><mi>b</mi><mo>+</mo><mi>y</mi></mtd><mtd><mi>c</mi><mo>+</mo><mi>z</mi></mtd></mtr><mtr><mtd><mi>a</mi><mo>+</mo><mi>p</mi></mtd><mtd><mi>b</mi><mo>+</mo><mi>q</mi></mtd><mtd><mi>c</mi><mo>+</mo><mi>r</mi></mtd></mtr></mtable></mfenced></math> then
Step-by-Step Solution
Key Concept: The determinant B can be simplified using row operations. Adding all rows gives 2(a+p+x), 2(b+q+y), 2(c+r+z). Factoring out 2 and performing row subtractions reveals B = 2 * |p+x q+y r+z; a+x b+y c+z; a+p b+q c+r| which simplifies to 2 * 2 * |a b c; p q r; x y z| = 2 * 2 * A = 4 * 6 = 24? Wait, let's re-evaluate: B = |p+x q+y r+z; a+x b+y c+z; a+p b+q c+r|. Let R1=R1+R2+R3, then R1 = 2(a+p+x, b+q+y, c+r+z). Then R2=R2-R1/2 = (-p, -q, -r) and R3=R3-R1/2 = (-x, -y, -z). This leads to B = 2 * |a+p+x b+q+y c+r+z; -p -q -r; -x -y -z|. Adding R2 and R3 to R1 gives 2 * |a b c; -p -q -r; -x -y -z| = 2 * (-1) * (-1) * |a b c; p q r; x y z| = 2 * A = 12.
Let A = |a b c; p q r; x y z| = 6. B = |p+x q+y r+z; a+x b+y c+z; a+p b+q c+r|. Applying R1 -> R1+R2+R3, we get B = |2(a+p+x) 2(b+q+y) 2(c+r+z); a+x b+y c+z; a+p b+q c+r| = 2 * |a+p+x b+q+y c+r+z; a+x b+y c+z; a+p b+q c+r|. Now R2 -> R2-R1 and R3 -> R3-R1 gives 2 * |a+p+x b+q+y c+r+z; -p -q -r; -x -y -z|. Finally R1 -> R1+R2+R3 gives 2 * |a b c; -p -q -r; -x -y -z| = 2 * (-1) * (-1) * |a b c; p q r; x y z| = 2 * A = 2 * 6 = 12.
Correct Answer: (C)