Special Types of Matrices

The value of $x$, for which all the eigenvalues of the matrix given below are real is:

$A=\left[\begin{array}{ccc}10& 5+j& 4\\ x& 20& 2\\ 4& 2& -10\end{array}\right]$

If the matrix $A=\left[\begin{array}{ccc}3-x& 2& 2\\ 2& 4-x& 1\\ -2& -4& -1-x\end{array}\right]$  be a singular and orthogonal matrix. Then the value of $x$ is

Let $\mathrm{Y}={\mathrm{A}}^{\mathrm{n}}$

Find number of possible triplets $\left(\alpha , \beta , \gamma \right)$, if $A=\left|\begin{array}{ccc}0& \alpha & \alpha \\ 2\beta & \beta & -\beta \\ \gamma & -\gamma & \gamma \end{array}\right|$ is an orthogonal matrix.

Cotyledons are also called-

If $\mathrm{A}$ is an orthogonal matrix, then $\left|\mathrm{A}\right|$ is

If $\mathrm{A}$ is an orthogonal matrix, then ${\mathrm{A}}^{-1}$ equals

If $\mathrm{A}=\left[\begin{array}{cc}\cos \mathrm{\theta }& -\sin \mathrm{\theta }\\ \sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right]$ , then

If $A \text{and} B$ are two matrices such that $AB=B \text{and} BA=A$, then

If $i=\sqrt{-1}, a=\frac{1+\sqrt{5}}{2}, b=\frac{1-\sqrt{5}}{2}$, then which of the following matrix is idempotent?

If $\mathrm{k}\mathrm{ }\left[\begin{array}{lll}-1& 2& 2\\ 2& -1& 2\\ 2& 2& -1\end{array}\right]$ is an orthogonal matrix then $\mathrm{k}$ is equal to

If $\mathrm{A}$ is idempotent and $\mathrm{A}+\mathrm{B}=\mathrm{I}$ , then which of the following true?

If the matrix $\left[\begin{array}{lll}0& 2\mathrm{\beta }& \mathrm{\gamma }\\ \mathrm{\alpha }& \mathrm{\beta }& -\mathrm{\gamma }\\ \mathrm{\alpha }& -\mathrm{\beta }& \mathrm{\gamma }\end{array}\right]$ is orthogonal, then

If $\mathrm{A}$ is an orthogonal matrix, then

If $\mathrm{k}\mathrm{ }.\mathrm{ }\left[\begin{array}{lll}-1& 2& 2\\ 2& -1& 2\\ 2& 2& -1\end{array}\right]$ is an orthogonal matrix, then k is equal to-

If $\mathrm{k}\left[\begin{array}{lll}-1& 2& 2\\ 2& -1& 2\\ 2& 2& -1\end{array}\right]$ is an orthogonal matrix then $\mathrm{k}$ is equal to-

For $\mathrm{k}\mathrm{ }=\frac{1}{\sqrt{50}}$ , the value of $\mathrm{a},\mathrm{ }\mathrm{b},\mathrm{ }\mathrm{c}$ such that $\mathrm{P}\mathrm{P}\mathrm{\prime }=\mathrm{I}$ , where $\mathrm{P}=\left[\begin{array}{lll}2/3& 3\mathrm{k}& \mathrm{a}\\ -1/3& -4\mathrm{k}& \mathrm{b}\\ 2/3& -5\mathrm{k}& \mathrm{c}\end{array}\right]$ is-

If the matrix $\left[\begin{array}{ccc}0& 2\beta & \gamma \\ \alpha & \beta & -\gamma \\ \alpha & -\beta & \gamma \end{array}\right]$ is orthogonal, then appropriate option will be

If $AB=I$ and $B=A^{\text{'}}$ then :

If $3\mathrm{A}=\left[\begin{array}{lll}1& 2& 2\\ 2& 1& –2\\ x& 2& y\end{array}\right]$ and $\mathrm{A}$ is orthogonal, then $x+y=$