Hermitian Matrix

Q.

Let $\omega =-\frac{1}{2}+i\frac{\sqrt{3}}{2}$, then the value of the determinant$\left|\begin{array}{ccc}1& 1& 1\\ 1& -1-{\omega }^2& {\omega }^2\\ 1& {\omega }^2& \omega \end{array}\right|$ is

1. $3\omega$

2. $3\omega (\omega -1)$

3. $3{\omega }^2$

4. $3\omega (1-\omega )$

Q. If $\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^3& b^3& c^3\end{array}\right|=(a-b)(b-c)(c-a)(a+b+c)$, then the determinant
$\left|\begin{array}{ccc}1& 1& 1\\ (x-a{)}^2& (x-b{)}^2& (x-c{)}^2\\ (x-b)(x-c)& (x-c)(x-a)& (x-a)(x-b)\end{array}\right|$
vanishes when

1. $a=-b=c$
2. $3x=a+b+c$
3. $a=b$ or $b=c$ or $c=a$
4. $3x=a-b+c$

Q.

Let $m,n\in N$ and $gcd(2,n)=1$. If $30{}^{30}C_0+29{}^{30}C_1+\dots ..+2{}^{30}C_{28}+1{}^{30}C_{29}=n.2^m$, then $n+m$ is equal to

Q.

Can Two Different Matrices have the Same Determinant?

Q.

The inverse of a matrix is defined for

1. all matrices

2. Only square matrices

3. Diagonal matrices

4. Rectangular matrices

Q. Consider $A$ and $B$ two square matrices of same order. Select the correct alternative

1. $|A+B|$ must be greater than $|A|$
2. If $AB=0$ either $A$ or $B$ must be zero matrix
3. $|AB|$ must be greater than $|A|$
4. $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ is not unit matrix

Q.

Which of the following represents the condition for a matrix A to be hermitian Matrix. Given that the general element of the matrix is aij.

1. = -aij

2. aij =

3. aij =

4. aij =

Q.

A Matrix of any order can be a hermitian matrix

1. True

2. False

Q. Which of the following value(s) of $\alpha$ satisfy the equation $\left|\begin{array}{ccc}(1+\alpha {)}^2& (1+2\alpha {)}^2& (1+3\alpha {)}^2\\ (2+\alpha {)}^2& (2+2\alpha {)}^2& (2+3\alpha {)}^2\\ (3+\alpha {)}^2& (3+2\alpha {)}^2& (3+3\alpha {)}^2\end{array}\right|=-648\alpha$

1. $9$
2. $-9$
3. $3$
4. $-3$

Q.

The following matrix is HERMATIAN Matrix $\left[\begin{array}{ccc}1& -i& 5\\ i& 5& 0\\ 5& 0& 3\end{array}\right]$

1. True

2. False

Q.

If ${}^6{\mathrm{P}}_{\mathrm{r}}=360$ and if ${}^6\mathrm{C}_{\mathrm{r}}=15$, find $\mathrm{r}$?

Q. The value of $\left|\begin{array}{ccc}a+x& y& z\\ x& a+y& z\\ x& y& a+z\end{array}\right|$ is equal to

1. $0$
2. $a^2(a+z+x+y)$
3. $a^2(z+x+y)$
4. $a\cdot (a+z+x+y)$

Q. Let $A$ and $B$ be two symmetric matrices of order 3.
Statement-1:
$A\left(BA\right)$ and $\left(AB\right)A$ are symmetric matrices.
Statement-2:
$AB$ is symmetric matrix if matrix multiplication of $A$ and $B$ is commutative.

1. Statement-1 is true, Statement-2 is true; Statement-2 is correct explanation for Statement-1.
2. Statement-1 is false, Statement-2 is true.
3. Statement-1 is true, Statement-2 is false
4. Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

Q.

The following matrix is HERMATIAN Matrix $\left[\begin{array}{ccc}1& -i& 5\\ i& 5& 0\\ 5& 0& 3\end{array}\right]$

1. True

2. False

Q. If A = [a_ij] is a 3 × 3 scalar matrix such that a₁₁ = 2, then write the value of |A|.

Q.

A Hermitian matrix is a square matrix with complex entries that is equal to its own conjugate.

1. True

2. False

Q.

1. 1 x 5

2. 1

3. 5

4. 5 x 1

Q. Give below are defined as few properties of matrices. Which of the following is not true for matrices?

1. $AB=AC$ does not imply $B=C$
2. $A+B=B+A$
3. $(AB{)}^{\prime }=B^{\prime }{A}^{\prime }$
4. $AB=0$ implies $|A|=0$ or $|B|=0$

Q. If $A$ is singular matrix, then $\text{adj}A$ is

1. Symmetric
2. Singular
3. Non-singular
4. Not defined

Q. If $f\left(x\right)=\left|\begin{array}{cc}6x^3& x^2\\ 2& x^6\end{array}\right|,$ then which of the following(s) is/are true?

1. $f^{\prime }\left(0\right)=1$
2. $f^{\prime }\left(0\right)=0$
3. $f^{\prime }\left(1\right)=100$
4. $f^{\prime }\left(1\right)=50$

Q.

Which of the following represents the condition for a matrix A to be hermitian Matrix. Given that the general element of the matrix is aij.

1. $aij=-\overline{aij}$

2. $aij=\overline{aji}$

3. $aij=-\overline{aji}$

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

Q. If $A^{-1}=\left[\begin{array}{ccc}1& -1& 2\\ 0& 3& 1\\ 0& 0& -1/3\end{array}\right],$ then

1. $|A|=-1$
2. $A=\left[\begin{array}{ccc}1& -1/3& -7\\ 0& -3& 0\\ 0& 0& 1\end{array}\right]$
3. $\text{adj}A=\left[\begin{array}{ccc}-1& 1& -2\\ 0& -3& -1\\ 0& 0& 1/3\end{array}\right]$
4. $A=\left[\begin{array}{ccc}1& 1/3& 7\\ 0& 1/3& 1\\ 0& 0& -3\end{array}\right]$