Symmetric Matrix

Q. Let $A$ and $B$ be $3\times 3$ real matrices such that $A$ is symmetric matrix and $B$ is skew-symmetric matrix. Then the system of linear equations $(A^2{B}^2-B^2{A}^2)X=O,$ where $X$ is a $3\times 1$ column matrix of unknown variables and $O$ is a $3\times 1$ null matrix, has

1. a unique solution
2. exactly two solutions
3. infinitely many solutions
4. no solution

Q.

Let ${\left(2x^2+3x+4\right)}^{10}=\sum _{r=0}^{20}{a}_r{x}^r$, then $\frac{a_7}{a_{13}}=$

Q. If $A$ is a symmetric matrix and $B$ is a skew- symmetrix matrix such that $A+B=\left[\begin{array}{cc}2& 3\\ 5& -1\end{array}\right],$ then $AB$ is equal to :

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

Q.

If $A$ is a skew symmetric matrix, then $A^2$ is a

Q. If A is a skew-symmetric matrix of order 3, then the matrix $A^4$ is

1. symmetric
2. skew symmetric
3. diagonal
4. none of those

Q.

Express the matrix $B=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$ as the sum of a symmetric and a skew symmetric matrix.

Q. Let $A=\left(\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ 0& 0& 1\end{array}\right)$ and $B=7A^{20}-20A^7+2I,$ where $I$ is an identity matrix of order $3\times 3$. If $B=\left[b_{ij}\right],$ then $b_{13}$ is equal to

Q. If $A$ is a square matrix, then

1. $A{A}^T$ is symmetric matrix and $A^{T}A$ is skew-symmetric matrix.
2. $A{A}^T$ is skew-symmetric matrix and $A^{T}A$ is symmetric matrix.
3. Both $A{A}^T$ and $A^{T}A$ are symmetric matrices.
4. Both $A{A}^T$ and $A^{T}A$ are skew-symmetric matrices.

Q. Let $A$ be the set of all $3\times 3$ symmetric matrices, all of whose entries are either $0$ or $1$, five of these are $1$ and four of them are $0$

Consider the system of linear equations $A\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$, then which of the following is/are correct?

1. The total number of possible matrices in $A$ is $12$.
2. The number of matrices $A$ in set $A$ for which the system of linear equations has a unique solution is $6$.
3. The number of matrices $A$ in set $A$ for which the system of linear equations has a unique solution is $4$.
4. The number of matrices $A$ in set $A$ for which the system of linear equations is inconsistent is more than $2$.

Q. Which among the following graph(s) is/are even function?

1. 
2. 
3. 
4. 

Q.

Show that the matrix B' AB is symmetric or skew-symmetric according to A which is symmetric or skew -symmetric.

Q.

A square matrix A is said to be a symmetric matrix if

1. $A=-A^T$

2. $A=A^T$

3. $A=\overline{A}$

4. $A=-\overline{A}$

Q. If $A=\left[\begin{array}{ccc}5& 2& a\\ b& 0& -3\\ 4& c& -7\end{array}\right]$ is a symmetric matrix, then find the value of $(a+b+c)$.

1. $3$
2. $11$
3. $15$
4. $2$

Q. If the matrix$\left[\begin{array}{ccc}0& a& 3\\ 2& b& -1\\ c& 1& 0\end{array}\right]$is a skew symmetric matrix, then the value of $|a+b+c|$ is

Q.

Is a zero matrix symmetric?

Q. If $A$ is a square matrix, then which of the following is correct ?

$\left(a\right)$ $A{A}^T$ is symmetric matrix and $A^{T}A$ is skew-symmetric matrix.
$\left(b\right)$ $A{A}^T$ is skew-symmetric matrix and $A^{T}A$ is symmetric matrix.
$\left(c\right)$ Both $A{A}^T$ and $A^{T}A$ are symmetric matrices.
$\left(d\right)$ Both $A{A}^T$ and $A^{T}A$ are skew-symmetric matrices.

Q. If A , B are symmetric matrices of same order, then AB − BA is a A. Skew symmetric matrix B. Symmetric matrix C. Zero matrix D. Identity matrix

Q. Let $ABC=I$ then $\text{tr}(ABC+BCA+CAB)$ is
(where order of matrices $A,B,C$ is $3$ and $\text{tr}\left(A\right)$ is sum of the principle diagonal elements in $A$)

Q. If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix.

Q.

What Are The Eigenvalues Of A Symmetric Matrix?

Q. For the matrix $A=\left[\begin{array}{cc}1& 5\\ 6& 7\end{array}\right],$ verify that
$\left(i\right)$ $(A+A^{\prime })$ is a symmetric matrix.
$\left(ii\right)$ $(A-A^{\prime })$ is a skew symmetric matrix.

Q.

In a skew-symmetric matrix, the diagonal elements are all.

1. One.

2. Zero.

3. Different from each other.

4. Non-zero.

Q. If A and B are two non singular matrices and both are symmetric and commute each other then

1. $A^{-1}B$ is symmetric but $A^{-1}{B}^{-1}$ is not symmetric
2. Both $A^{-1}B$ and $A^{-1}{B}^{-1}$ are symmetric
3. Neither $A^{-1}B$ nor $A^{-1}{B}^{-1}$ are symmetric
4. $A^{-1}{B}^{-1}$ is symmetric but $A^{-1}B$ is not symmetric

Q.

Express the following matrix as the sum of a symmetric and a skew-symmetric matrices;

$\left[\begin{array}{ccc}6& -2& 2\\ -2& 3& -1\\ 2& -1& 3\end{array}\right]$

Q. Distinct prime numbers $p,q,r$ satisfy the equation $2pqr+50pq=7pqr+55pr=8pqr+12qr=A,$ for some positive integer $A.$ Then $A$ is

1. $660$
2. $1980$
3. $388$
4. $180$

Q.

Express the following matrix as the sum of a symmetric and a skew-symmetric matrices;

$\left[\begin{array}{cc}1& 5\\ -1& 2\end{array}\right]$

Q. If $A$ and $B$ are non-singular matrix of order $3$ and $(I-AB)$ is invertible, then which of the following statements is/are correct:

1. $I-BA$ is not invertible
2. $I-BA$ is invertible
3. inverse of $(I-BA)$ is $I+B(I-AB{)}^{-1}A$
4. inverse of $(I-BA)$ is $I+A(I-BA{)}^{-1}B$

Q.

Two matrices are multiplied by multiplying their corresponding elements.

1. False

2. True

Q. Show that the matrix is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Q. Let $P$ be a non-singular matrix and $I+P+P^2+\cdots +P^n=O$. Then $P^{-1}$ is equal to (where $n\in Z^{+}$)

1. $P^n$
2. $P$
3. $P^{n-1}$
4. $I$