Singular and Non Singualar Matrices

Q. If
$A=\left[\begin{array}{cccc}{e}^t& e^{-t}\cos t& e^{-t}\sin t& \\ e^t& -e^{-t}\cos t-e^{-t}\sin t& -e^{-t}\sin t+e^{-t}\cos t& \\ e^t& 2e^{-t}\sin t& -2e^{-t}\cos t& \end{array}\right]$,

then $A$ is :

1. invertible only if $t=\pi$.
2. invertible only if $t=\frac{\pi }{2}$.
3. invertible for all $t\in R$.
4. not invertible for any $t\in R$.

Q.

$IfmatrixA=[a_{ij}{]}_{3\times 3},matrixB=[b_{ij}{]}_{3\times 3}where{a}_{ij}+a_{ji}=0and{b}_{ij}-b_{ji}=0,then|A^4.B^3|is$

1. skew-symmetric matrix

2. singular

3. symmetric

4. zero matrix

Q.

A matrix ‘B’ is singular if

1. B is invertible

2. |B| = 0

3. $|B|\ne 0$

4. An inverse of B doesn’t exist

Q.

A matrix ‘B’ is singular if

1. B is invertible

2. |B| = 0

3. $|B|\ne 0$

4. An inverse of B doesn’t exist

Q. Let $P$ and $Q$ be two matrices different from identity matrix $I$ and null matrix $O$ such that $PQ=QP$ and if $P^6-Q^6=P^5-Q^5=P^4-Q^4=I$ then,

1. $I-P$ is singular
2. $I-Q$ is singular
3. $P+Q=PQ$
4. $(I-P)(I-Q)$ is non singular

Q. Let $A$ be a symmetric matrix such that $A^4=0$ and $B=I+A+A^2+A^3,$ then $B$ is

1. singular matrix
2. symmetric matrix
3. non-singular matrix
4. skew symmetric matrix

Q. $\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{a. If A is an indempotent matrix and I is an}& \\ \text{identity matrix of the same order, then the}& \\ \text{value of}n,\text{such that}(A+I{)}^n=I+127A\text{is}& p.9\\ \text{b. If}(I-A{)}^{-1}=I+A+A^2+....+A^7,& \\ \text{then}{A}^n=0\text{where}n\text{is}& \text{q.}10\\ \text{c. If A is matrix such that}{a}_{ij}=(i+j)(i-j),& \\ \text{then A is singular if order of marix is}& r.7\\ \text{d. If a non-singular matrix A is symmetric,}& \\ \text{show that}{A}^{-1}\text{is also symmetric, then order}& \\ \text{of A can be}& s.8\end{array}$
Which of the following options is/are correct?

1. $a\to r;b\to s;c\to ,r;d\to p$
2. $a\to r;b\to p,q,s;c\to ,p,r;d\to p,q,r,s.$
3. $a\to r;b\to p,,s;c\to ,p,r;d\to r,s.$
4. $a\to s;b\to p,q,s;c\to ,p,r;d\to q,r,s.$

Q. Let P and Q be $3\times 3$ matrices such that $P\ne Q.$ If $P^3=Q^{3}and{P}^{2}Q=Q^{2}P,$ then determinant of $(P^2+Q^2)$ is equal to:

1. -2
2. 1
3. \N
4. -1