Let A=[x&y&z y&z&x z&x&y] where x, y and z are real numbers such that x+y+z0 and x y z=2 If A2=I3, then the value of x3+y3+z3 is

Question

Let A=[x&y&z y&z&x z&x&y] where x, y and z are real numbers such that x+y+z>0 and x y z=2 If A2=I3, then the value of x3+y3+z3 is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

A 2= I ⇒ AA'= I (. as . A prime= A ) ⇒ A is orthogonal So, x2+y2+z2=1 and x y+y z+z x=0 ⇒ (x+y+z)2=1+2 × 0 ⇒ x+y+z=1 Thus, x 3+ y 3+ z 3 =3 × 2+1 ×(1-0) =7

Q. Let $A = ⎣ ⎡ x y z y z x z x y ⎦ ⎤$ where $x, y$ and $z$ are real numbers such that $x + y + z > 0$ and $x yz = 2$ If $A^{2} = I_{3}$ , then the value of $x^{3} + y^{3} + z^{3}$ is

$A^{2} = I$
$\Rightarrow A A^{'} = I$
$($ as $A^{'} = A)$
$\Rightarrow A$ is orthogonal
So, $x^{2} + y^{2} + z^{2} = 1$ and $x y + yz + z x = 0$
$\Rightarrow (x + y + z)^{2} = 1 + 2 \times 0$
$\Rightarrow x + y + z = 1$
Thus,
$x^{3} + y^{3} + z^{3} = 3 \times 2 + 1 \times (1 - 0)$
$= 7$

Q. Let A=⎣⎡​xyz​yzx​zxy​⎦⎤​ where x,y and z are real numbers such that x+y+z>0 and xyz=2 If A2=I3​, then the value of x3+y3+z3 is

A2=I ⇒AA′=I ( as A′=A) ⇒A is orthogonal So, x2+y2+z2=1 and xy+yz+zx=0 ⇒(x+y+z)2=1+2×0 ⇒x+y+z=1 Thus, x3+y3+z3=3×2+1×(1−0) =7

Q. Let $A = ⎣ ⎡ x y z y z x z x y ⎦ ⎤$ where $x, y$ and $z$ are real numbers such that $x + y + z > 0$ and $x yz = 2$ If $A^{2} = I_{3}$ , then the value of $x^{3} + y^{3} + z^{3}$ is

$A^{2} = I$
$\Rightarrow A A^{'} = I$
$($ as $A^{'} = A)$
$\Rightarrow A$ is orthogonal
So, $x^{2} + y^{2} + z^{2} = 1$ and $x y + yz + z x = 0$
$\Rightarrow (x + y + z)^{2} = 1 + 2 \times 0$
$\Rightarrow x + y + z = 1$
Thus,
$x^{3} + y^{3} + z^{3} = 3 \times 2 + 1 \times (1 - 0)$
$= 7$