Skew Symmetric Matrix

Q.

If A is any square matrix, then AA’ is a

1. Skew- symmetric matrix

2. Symmetric matrix

3. Hatmitian Matrix

4. Skew-harmitian Matrix

Q.

What is the difference between ${}^{n}P_r$ and ${}^{n}C_r$?

Q.

A matrix which is both symmetric as well as skew-symmetric is a null matrix. Prove.

Q. Matrices of order $3\times 3$ are formed using the elements of set $A=\{-3,-2,-1,0,1,2,3\}.$ Then the probability that matrices are either symmetric or skew-symmetric, is

1. $\frac{1}{7^6}+\frac{1}{7^3}-\frac{1}{7^8}$
2. $\frac{1}{7^3}+\frac{1}{7^6}-\frac{1}{7^9}$
3. $\frac{1}{7^6}+\frac{1}{7^3}$
4. $\frac{1}{7^3}+\frac{1}{7^8}$

Q.

Evaluate the determinant $\Delta =\left|\begin{array}{ccc}1& 2& 4\\ -1& 3& 0\\ 4& 1& 0\end{array}\right|$

Q. Let $A$ and $B$ be two square matrices of order $3.$ Then which of the following statements is(are) CORRECT?

1. $AB{A}^T$ is symmetric matrix.
2. $AB-BA$ is skew symmetric matrix.
3. If $B=|A|A^{-1},$ $|A|\ne 0,$ then $adj\left(A^T\right)-B$ is skew symmetric matrix.
4. If $B+A^T=O$ and $A$ is skew symmetric matrix, then $B^{15}$ is also skew symmetric matrix.

Q. If A and B are skew symmetric matrices then (AB - BA) is a matrix and (AB + BA) is a matrix.

1. skew - symmetric
2. symmetric matrix
3. null matrix
4. identity matrix

Q. Let $X$ and $Y$ be two arbitrary, $3\times 3$, non-zero, skew symmetric matrices and $Z$ be an arbitrary $3\times 3$, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

1. $Y^3{Z}^4-Z^4{Y}^3$
2. $X^{44}+Y^{44}$
3. $X^4{Z}^3-Z^3{X}^4$
4. $X^{23}+Y^{23}$

Q. If R and R` are symmetric relations (not disjoint) on a set A, then the relation R union R` is

Q. If $P$ and $Q$ are non singular symmetric matrices and $PQ=QP,$ then

1. $Q{P}^{-1}\ne P^{-1}Q$
2. $P^{-1}Q$ is symmetric.
3. $Q^{-1}P\ne P{Q}^{-1}$
4. $P{Q}^{-1}$ is non-symmetric.

Q. For what value of x, is the matrix $A=\left[\begin{array}{ccc}0& 1& -2\\ -1& 0& 3\\ x& -3& 0\end{array}\right]$ a skew-symmetric matrix?

Q.

If $A$ is a symmetric matrix and $n\in N$, then $A^n$ is

1. symmetric matrix

2. a diagonal matrix

3. a skew-symmetric

4. None of the above

Q.

If $\mathrm{A}$ is a symmetric and $\mathrm{B}$ skew-symmetric matrix and $\mathrm{A}+\mathrm{B}$ is non-singular and $\mathrm{C}={(\mathrm{A}+\mathrm{B})}^{-1}(\mathrm{A}-\mathrm{B})$, then prove that

$$\begin{gathered} \left(\mathrm{i}\right){\mathrm{C}}^{\mathrm{T}}(\mathrm{A}+\mathrm{B})\mathrm{C}=\mathrm{A}+\mathrm{B} \\ \left(\mathrm{ii}\right){\mathrm{C}}^{\mathrm{T}}(\mathrm{A}-\mathrm{B})\mathrm{C}=\mathrm{A}-\mathrm{B} \\ \left(\mathrm{iii}\right){\mathrm{C}}^{\mathrm{T}}\mathrm{AC}=\mathrm{A} \end{gathered}$$

Q. If A is a skew symmetric matrix of odd order n then prove that |A|=0

Q. Let $A=\left(\begin{array}{ccc}12& 24& 5\\ x& 6& 2\\ -1& -2& 3\end{array}\right)$. The value of $x$ for which the matrix $A$ is not invertible is

1. $6$
2. $12$
3. $3$
4. $2$

Q. Write a 2 × 2 matrix which is both symmetric and skew-symmetric.

Q.

For the matrix, verify that

(i) is a symmetric matrix

(ii) is a skew symmetric matrix

Q. If both $(A-\frac{I}{2})$ and $(A+\frac{I}{2})$ are orthogonal matrices, then

1. A is skew symmetric matrix of even order
2. $A^2=\frac{3}{4}I$
3. A is orthogonal
4. A is skew symmetric of odd order

Q.

Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of $A^2$ is $1$, then the possible number of such matrices is :

1. $6$

2. $1$

3. $4$

4. $12$

Q. If matrix $A=\left[\begin{array}{cc}2x& -4\\ -2& x\end{array}\right]$ is singular, then the value(s) of $x$ can be

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

Q. Assertion $\left(A\right)$:
The matrix $A=\left[\begin{array}{ccc}0& a& b\\ -a& 0& c\\ -b& -c& 0\end{array}\right]$ is a skew-symmetric matrix.
Reason $\left(R\right)$:
A square matrix $A=a_{ij}$ of order $m$ is said to be skew symmetric if $A^T=-A$.

1. Both $\left(A\right)$ and $\left(R\right)$ individually true and $\left(R\right)$ is correct explanation of $\left(A\right)$
2. Both $\left(A\right)$ and $\left(R\right)$ individually true but $\left(R\right)$ is not the correct explanation of $\left(A\right)$
3. $\left(A\right)$ is true but $\left(R\right)$ is false
4. $\left(A\right)$ is false but $\left(R\right)$ is true

Q. Let $A$ and $B$ be two square matrices satisfying $AB-BA=A$. Then which of the following is (are) TRUE?

1. If $A$ is non-singular, then $B=O$, where $O$ is a null matrix.
2. If $A$ is non-singular, then $\text{det}(B-I)=\text{det}(B+I)$, where $I$ is an identity matrix.
3. If $A$ and $B$ are symmetric matrices of order $3$, then $A$ is singular.
4. If $A$ and $B$ are skew-symmetric matrices of order $3$, then $A$ is non-singular.

Q. If A and B are matrices of the same order, then AB^T − BA^T is a

(a) skew-symmetric matrix
(b) null matrix
(c) unit matrix
(d) symmetric matrix

Q. Assertion : The determinant of a skew symmetric matrix of even order is perfect square.
Reason : The determinant of a skew symmetric matrix of odd order is equal to zero.

1. Assertion and reason both are correct and reason is correct explanation of the assertion.
2. Assertion is wrong.
3. Assertion and reason both are correct but reason is not correct explanation of the assertion.
4. Reason is wrong.

Q. The number of $4\times 4$ skew symmetric matrices that can be formed using exactly four $0$'s, six $3$'s and six $-3$'s is

Q.

Let $A$ be the set of all $3\times 3$ skew-symmetric matrices whose entries are either $-1,0,$ or $1$. If there are exactly three $0s$, three $1s$, and three $(-1)s$, then what is the number of such matrices?

Q.

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

(i)

(ii)

(iii)

(iv)

Q. $A$ and $B$ are square matrices of order $3\times 3$, $A$ is an orthogonal matrix and $B$ is a skew symmetric matrix. Which of the following statement is not true.

1. $|A|=\pm 1$
2. $|B|=0$
3. $|AB|=1$
4. $|AB|=0$

Q. The determinant of an odd order skew symmetric matrix is always :

1. One
2. Zero
3. Negative
4. Depends on the matrix

Q. If $A$ is a $3\times 3$ matrix such that $|5\cdot adjA|=5,$ then $|A|$ is equal to

1. $\pm \frac{1}{5}$
2. $\pm 5$
3. $\pm 1$
4. $\pm \frac{1}{25}$