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JEE Main · 2021 · 25 Feb 2021 · Shift 1 · Matrices

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

A^{2} = I_{3}

, then the value of

x^{3} + y^{3} + z^{3}

is ____

Answer: 7

Solution

The matrix is symmetric, and the first-row entries of

A^{2}

are

(A^{2})_{11} = x^{2} + y^{2} + z^{2}, (A^{2})_{12} = x y + y z + z x .

So

A^{2} = I_{3}

forces

x^{2} + y^{2} + z^{2} = 1

and

x y + y z + z x = 0

. Then

(x + y + z)^{2} = 1 + 2 \cdot 0 = 1

, and since

x + y + z > 0

,

x + y + z = 1

. Using

x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x) = 1 \cdot 1 = 1

together with

x y z = 2

, the value of

x^{3} + y^{3} + z^{3}

is

1 + 6 = 7

.

JEE Main · 2021 · 25 Feb 2021 · Shift 1 · Integer answer

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