Orthogonal Matrix

Q. If a matrix $A$ is both symmetric and skew symmetric, then

1. $A$ is a diagonal matrix
2. $A$ is a zero matrix
3. $A$ is a scalar matrix
4. $A$ is a square matrix

Q.

If A and B are invertible matrices, then which of the following is not correct?

(a) $adjA=|A|.A^{-1}$
(b) $det(A{)}^{-1}=[det\left(A\right){]}^{-1}$
(c) $(AB{)}^{-1}=B^{-1}{A}^{-1}$
(d) $(A+B{)}^{-1}=B^{-1}+A^{-1}$

Q. Let $A=\left(\begin{array}{ccc}0& 2q& r\\ p& q& -r\\ p& -q& r\end{array}\right)$. If $A{A}^T=I_3$, then $|p|$ is:

1. $\frac{1}{\sqrt{2}}$
2. $\frac{1}{\sqrt{3}}$
3. $\frac{1}{\sqrt{5}}$
4. $\frac{1}{\sqrt{6}}$

Q. Let $A$ and $B$ be any two $3\times 3$ symmetric and skew symmetric matrices respectively. Then Which of the following is NOT true?

Q. If A is an orthogonal matrix and $|A|=-1,$ then $A^T$ is equal to

1. $-A$
2. $A$
3. $-\text{adj}\left(A\right)$
4. $\text{adj}\left(A\right)$

Q. Let $A$ and $B$ be two $n\times n$ real symmetric matrices. Which of the following is/are correct?

1. If $A$ is an involuntry matrix then, $A^{-1}=A$
2. If $A$ is an orthogonal matrix then, $A^{-1}=A$
3. The product $AB$ is symmetric if $AB=BA$
4. If the product $AB$ is diagonal then, $AB=BA$

Q.

$A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& \frac{-1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]$is an orthogonal matrix

1. True

2. False

Q. If $S=\left[\begin{array}{cc}6& -8\\ 2& 10\end{array}\right]=P+Q$, where $P$ is a symmetric & $Q$ is a skew -symmetric matrix, then $Q$ is

1. $\left[\begin{array}{cc}0& 5\\ -5& 0\end{array}\right]$
2. $\left[\begin{array}{cc}0& -5\\ 5& 0\end{array}\right]$
3. $\left[\begin{array}{cc}0& 8\\ -8& 0\end{array}\right]$
4. $\left[\begin{array}{cc}0& 6\\ -6& 0\end{array}\right]$

Q. Let $A=\left(\begin{array}{cc}i& 0\\ 0& i\end{array}\right),B=\left(\begin{array}{cc}-i& 0\\ 0& -i\end{array}\right),i=\sqrt{-1}$
$\text{Statement-I}:(A+B)(A-B)=A^2-B^2\text{Statement-II}:AB=BA$
Then which of the following is correct regarding the given statements:

1. Statement $1$ is false. statement $2$ is true.
2. Statement $1$ is true. statement $2$ is true ; statement $2$ is correct explanation for statement $1$
3. Statement $1$ is true. statement $2$ is true ; statement $2$ is not a correct explanation for statment $1$.
4. Statement $1$ is true. statement $2$ is false.

Q. The number of distinct $5\times 5$ symmetric matrices with each element being $0$ or $1$ is

1. $2^{25}$
2. ${25}^2$
3. ${15}^2$
4. $2^{15}$

Q.
If $A$ is a non singular orthogonal matrix and $B=A^{-1}{A}^T$, then matrix $B$ is

1. An involutory matrix​​​​​
2. An idempotent matrix
3. An orthogonal matrix
4. None of the above

Q. If both $A-\frac{1}{2}I$ and $A+\frac{1}{2}I$ are othogonal matrices, then

1. $A$ is orthogonal
2. $A$ is skew-symmetric matrix of even order
3. $A^2=\frac{3}{4}I$
4. none of these

Q. If A is an orthogonal matrix of order n, then the value of |adj.(adj A)| is

1. $n$
2. $\pm 1$
3. $0$
4. $n-2$

Q. Let $A=\left[\begin{array}{ccc}X& Y& Z\end{array}\right]$ be a $3\times 3$ orthogonal matrix with $X,Y,Z$ as its column vectors. Then $B=X{X}^T+Y{Y}^T$

1. is a symmetric matrix
2. is the $3\times 3$ identity matrix
3. satisfies $B^2=B$
4. satisfies $BZ=O,O$ being the null matrix

Q. If $A$ is an orthogonal matrix and $B=AB,$ then $B^{T}A\left(B^T\text{is transpose of}B\right)$ is

1. $A$
2. $B$
3. $A^T$
4. $B^T$

Q. Let $M=\left[\begin{array}{ccc}x& 2x& 3x\\ f\left(x\right)& g\left(x\right)& h\left(x\right)\\ 0& 1& 1\end{array}\right],x\ne 0$ be a singular matrix. If $f\left(x\right)=\ln (e^x+1)$ and $g\left(x\right)=\ln (e^x-1),$ then the value of $h^{\prime }\left(\ln 3\right)$ is

1. $\frac{9}{8}$
2. $\frac{9}{4}$
3. $3$
4. $6$

Q.

Find the literal factor and numerical factor from the following.

$-26xy{z}^3$

Q. Construct a $2\times 2$ matrix, $A=\left[a_{ij}\right]$, whose elements are given by:
(i) $a_{ij}=\frac{(i+j{)}^2}{2}$
(ii) $a_{ij}=\frac{i}{j}$
(iii) $a_{ij}=\frac{(i+2j{)}^2}{2}$

Q.

If A is a $(3\times 3)$ skew symmetric matrix, formed by using the digits 0, 1, -1, then the number of such matrices is

1. 10
2. 27
3. 8
4. 15

Q.

Using properties of determinants prove the following questions.

$\left|\begin{array}{ccc}1& 1+p& 1+p+q\\ 2& 3+2p& 4+3p+2q\\ 3& 6+3p& 10+6p+3q\end{array}\right|=1$

Q. If $a\ne p,b\ne q,c\ne r$ and $\left|\begin{array}{ccc}p& b& c\\ p+a& q+b& 2c\\ a& b& r\end{array}\right|=0$, then $\frac{p}{p-q}+\frac{q}{q-b}+\frac{r}{r-c}=$

1. 3
2. 2
3. 1
4. 0

Q. The function $f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}ax(x-1)+b& x<1\end{array}\\ \begin{array}{cc}x-1& 1\le x\le 3\end{array}\\ \begin{array}{cc}p{x}^2+qx+2& x>3\end{array}\end{array}$

1. $p=\frac{1}{3}$
2. $a\in R-\left\{1\right\}$
3. $b=0$
4. $q=-1$

Q.

$A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& \frac{-1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]$is an orthogonal matrix

1. True

2. False

Q. If $x=a^2-bc,y=b^2-ca,z=c^2+ab$, then value of $\frac{(a+b+c)(x+y+z)}{ax+by+cz}$ is equal to:

1. $1$
2. $3$
3. $2$
4. $0$

Q. Let $A=\left(\begin{array}{ccc}0& 2q& r\\ p& q& -r\\ p& -q& r\end{array}\right)$. If $A{A}^T=I_3$, then $|p|$ is:

1. $\frac{1}{\sqrt{2}}$
2. $\frac{1}{\sqrt{3}}$
3. $\frac{1}{\sqrt{5}}$
4. $\frac{1}{\sqrt{6}}$

Q. If $A$ is a non-singular matrix of order $3\times 3$, then adj $\left(adjA\right)$ is equal to

1. $\left|A\right|A$
2. ${\left|A\right|}^{2}A$
3. ${\left|A\right|}^{-1}A$
4. none of these

Q. If $p{\lambda }^4+q{\lambda }^3+r{\lambda }^2+s\lambda +t=\left|\begin{array}{ccc}{\lambda }^2+3\lambda & \lambda -1& \lambda +3\\ {\lambda }^2+1& 2+5\lambda & \lambda -3\\ {\lambda }^2-3& \lambda +4& 3\lambda \end{array}\right|$
then $p$ is equal to

1. 5
2. 8
3. 3
4. 2

Q. Let $k$ be a positve real number and let
$A=\left[\begin{array}{ccc}2k-1& 2\sqrt{k}& 2\sqrt{k}\\ 2\sqrt{k}& 1& -2k\\ -2\sqrt{k}& 2k& -1\end{array}\right]$ and $B=\left[\begin{array}{ccc}-& 22k-1& 2\sqrt{k}\\ 1-2k& 0& 2\sqrt{k}\\ -\sqrt{k}& -2\sqrt{k}& 0\end{array}\right]$. If $\text{det (adj A)+det(adj B)}={10}^6$, then $\left[k\right]$ is equal to

$[$Note: $\text{adj M}$ denotes the adjoint of a square matrix $M$ and $\left[k\right]$ denotes the largest integer less than or equal to $k]$

Q.

Using the property of determinants and without expanding.

$\left|\begin{array}{ccc}b+c& q+r& y+x\\ c+a& r+p& z+x\\ a+b& p+q& x+y\end{array}\right|=2\left|\begin{array}{ccc}a& p& x\\ b& q& y\\ c& r& z\end{array}\right|$

Q. Let $A$ and $B$ be $3\times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skew-symmetric and $\left(A+B\right)\left(A-B\right)=\left(A-B\right)\left(A+B\right).$ If ${\left(AB\right)}^T={\left(-1\right)}^{k}AB,$ where ${\left(AB\right)}^T$ is the transpose of the matrix $AB,$ then the value of $k$ is:

1. $0$
2. $2$
3. $1$
4. $3$