Let A = begin{bmatrix} x & y & z y & z & x | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given $A = \begin{bmatrix} x & y & z \ y & z & x \ z & x & y \end{bmatrix}$. Since $A$ is symmetric,$A^T = A$.Given $A^2 = I$,we have $AA^T = I$

Vedclass · Accepted Answer

$7$

Vedclass · Answer

(A) Given $A = \begin{bmatrix} x & y & z \ y & z & x \ z & x & y \end{bmatrix}$. Since $A$ is symmetric,$A^T = A$.Given $A^2 = I$,we have $AA^T = I$,which means $A$ is an orthogonal matrix.For the matrix $A$,the condition $AA^T = I$ implies:$x^2 + y^2 + z^2 = 1$ and $xy + yz + zx = 0$.We know that $(x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)$.Substituting the values,$(x + y + z)^2 = 1 + 2(0) = 1$.Since $x + y + z > 0$,we have $x + y + z = 1$.Using the algebraic identity $x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - (xy + yz + zx))$,$x^3 + y^3 + z^3 - 3(2) = (1)(1 - 0)$.$x^3 + y^3 + z^3 - 6 = 1$.$x^3 + y^3 + z^3 = 7$.

Let $A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}$,where $x, y$ and $z$ are real numbers such that $x + y + z > 0$ and $xyz = 2$. If $A^2 = I_3$,then the value of $x^3 + y^3 + z^3$ is ............

Similar Questions

If $\frac{x^2+7}{(x^2+1)(x-2)}=\frac{A}{x-2}+\frac{Bx+C}{x^2+1}$, then the determinant of the matrix $\begin{bmatrix} A & B \\ C & \frac{2}{5} \end{bmatrix}$ is

Which of the following is(are) $NOT$ the square of a $3 \times 3$ matrix with real entries?

If one of the cube roots of $1$ be $\omega$, then $\left|\begin{array}{ccc}1 & 1+\omega^2 & \omega^2 \\ 1-i & -1 & \omega^2-1 \\ -i & -1+\omega & -1\end{array}\right|=$

If $A$ and $B$ are $3 \times 3$ matrices and $|A| \neq 0$,then which of the following are true?

Let $A = \begin{bmatrix} 1+i & 1 \\ -i & 0 \end{bmatrix}$ where $i = \sqrt{-1}$. Then,the number of elements in the set $\{n \in \{1, 2, \ldots, 100\} : A^n = A\}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module