Singular and Non Singular Matrices

Q. If $A$ and $B$ are symmetric matrices, then $ABA$ is

1. Symmetric matrix
2. Skew-symmetric
3. Diagonal matrix
4. Scalar matrix

Q.

Can we say that a zero matrix is invertible?

Q. A square matrix $A$ is skew-symmetric, such that $A^2+I=O,$ then

1. $A$ is orthogonal matrix
2. $A$ is orthogonal matrix of odd order
3. $A$ is orthogonal matrix of even order
4. $A$ is involutry matrix

Q.

Which among the following are singular matrix /matrices?

1. $\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right]$

2. $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$

3. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

4. $\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$

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. 0
4. -1

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. Match the columns.

Column - I	Column - II
(P) If $A=[a_{ij}{]}_{3\times 3}\text{and}{a}_{ij}=i^2+j^2,$ then A is	(p) Singular but not skew symmetric
(Q) $A=[a_{ij}{]}_{3\times 3}\text{and}{a}_{ij}=3^{i-j},$ then A is	(q) Skew-Symmetric
(R) $A=[a_{ij}{]}_{3\times 3}\text{and}{a}_{ij}=i^2-j^2,$ then A is	(r) Symmetric
(S) $A=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$	(s) An idempodent matrix which is singular

1. P Q R S
   3 4 2 1
2. P Q R S
   3 1 2 4
   
3. P Q R S
   3 1 4 2
4. P Q R S
   3 4 2 4

Q. If $A$ is a symmetric and $B$ is skew-symmetric matrix and $(A+B)$ is non-singular and $C=(A+B{)}^{-1}(A-B)$, then which of the following options is/are correct

1. $C^T(A+B)C=A+B$
2. $C^T(A+B)C=A-B$
3. $C^T(A-B)C=A-B$
4. $C^T(A-B)C=A+B$

Q. If$A=(a_{ij}{)}_{3\times 3}$ is a skew symmetric matrix, then

1. $a_{ii}\ne 0\forall i$
2. $A+A^T$ is a null matrix
3. $A^n$ is symmetric if ${}^{\prime }{n}^{\prime }$ is even natural number.
4. $A^n$ is skew-symmetric if ${}^{\prime }{n}^{\prime }$ is odd natural number.

Q. The circle $S_1$ has centre at $(1,2)$ and radius $3$; the circle $S_2$ has centre at $(9,8)$ and radius $7$. The circles $S_1$ and $S_2$ touch at the point whose co-ordinates are:

1. $\left(\begin{array}{c}\frac{17}{5},\frac{19}{5}\end{array}\right)$
2. $\left(\begin{array}{c}\frac{33}{5},\frac{31}{5}\end{array}\right)$
3. $\left(\begin{array}{c}\frac{17}{10},\frac{19}{10}\end{array}\right)$
4. $\left(\begin{array}{c}\frac{33}{10},\frac{31}{10}\end{array}\right)$

Q. A skew-symmetric matrix $A$ is such that $A^2+I=O,$ then which of the following is/are correct

1. $A$ is orthogonal matrix.
2. Order of $A$ is even.
3. Order of $A$ is odd.
4. $A$ is involutory matrix.