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

A matrix $B$ of order $n\times n$ is of the form $\lambda A$, where $\lambda$ is a scalar and $A$ has unit elements every where except in the main diagonal which has elements $\mu$. Then the values of $\lambda$ and $\mu$ so that $B$ will be orthogonal are

If matrix $A=\left[\begin{array}{lll}a& b& c\\ b& c& a\\ c& a& b\end{array}\right]$, where $a,b$ and $c$ are real positive number, $abc=1$ and $A^{T}A=I$, then which of the following is true

If $A=\left[\begin{array}{lll}2& 5& 3\\ 3& 1& 2\\ 1& 2& 1\end{array}\right]$ and $A^{-1}=\left[\begin{array}{ccc}-\frac{3}{4}& a& \frac{7}{4}\\ -\frac{1}{4}& b& \frac{5}{4}\\ \frac{5}{4}& c& -\frac{13}{4}\end{array}\right],$ then $a+b+c$ is equal to

Let $A$ be a square matrix such that $a_{ij}\in \left\{-1,0,1\right\} \forall i,j$ and it has only one non-zero entry in each row as well as in each column, then

If $\mathrm{A}$ is an idempotent matrix satisfying $(\mathrm{I} -0.4\mathrm{A}{)}^{-1}=\left(\mathrm{I}-\mathrm{\alpha A}\right)$ (where I is the unit matrix of same order as that of $\mathrm{A},\mathrm{ } \mathrm{A}$ is not a null matrix), then $\left|\frac{2}{\mathrm{\alpha }}\right|$ is

If $\mathrm{A}$ is an idempotent matrix satisfying $(\mathrm{I}-0.4\mathrm{A}{)}^{-1}=(\mathrm{I}-\alpha \mathrm{A})$ (where $\mathrm{I}$ is the unit matrix of same order as that of $\mathrm{A}, \mathrm{A}$ is not a null matrix), then $4\left|\frac{1}{\alpha }\right|$ 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.

If the matrix $\left(\begin{array}{ccc}a& \frac{2}{3}& \frac{2}{3}\\ \frac{2}{3}& \frac{1}{3}& b\\ c& x& y\end{array}\right)$ is an orthogonal matrix , then $3a-6b =$ 

Matrix $A=\left[\begin{array}{ccc}0& 2b& c\\ a& b& -c\\ a& -b& c\end{array}\right]$ is an orthogonal matrix, then find $|a b c|$ ?

Consider the following matrices $A=\left[\begin{array}{rrr}2& -3& -5\\ -1& 4& 5\\ 1& -3& -4\end{array}\right], B=\left[\begin{array}{rrr}1& -3& -4\\ -1& 3& 4\\ 1& -3& -4\end{array}\right] \text{and} C=\left[\begin{array}{rrr}0& 1& -1\\ 4& -3& 4\\ 3& -3& 4\end{array}\right]$, then matrix $A$ can be also written as 

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?