Transpose of a Matrix

Q.

$(AB{)}^T=A^T{B}^T$. The statement is

1. True

2. False

Q.

If $\theta$ is the exterior angle of a regular polygon of n sides and $\alpha$ is constant, then find the value of sin $\alpha$ + sin $(\alpha +\theta )$ + sin $(\alpha +2\theta )$ . . . . . up to n terms

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Q. If $A=\left[\begin{array}{ccc}1& 0& 1\\ 0& 2& 0\\ 1& -1& 4\end{array}\right],A=B+C,B=B^T,C=-C^T$, then $C$ is equal to

1. $\left[\begin{array}{ccc}0& 0.5& 0\\ -0.5& 0& 0\\ 0& 0& 0\end{array}\right]$
2. $\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0.5\\ 0& -0.5& 0\end{array}\right]$
3. $\left[\begin{array}{ccc}0& 0.5& 0\\ -0.5& 0& 0.5\\ 0& -0.5& 0\end{array}\right]$
4. $\left[\begin{array}{ccc}0& -0.5& 0.5\\ 0.5& 0& 0.5\\ -05& 0& 0\end{array}\right]$

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 involutory matrix
4. $A$ is orthogonal matrix of even order

Q. More than One Answer Type
एक से अधिक उत्तर प्रकार के प्रश्न

$z=\frac{3+4i}{3-4i}$, then

$z=\frac{3+4i}{3-4i}$, तब

1. $z=\frac{-7+24i}{25}$
2. |z| = 1
3. $z=\frac{7-24i}{25}$
4. |z| = 5

Q. Which among the following would be obtained on simplifying the expression $(A^{T}BA{)}^T$.

1. $A^T{B}^T{A}^T$
2. $A^T{B}^{T}A$
3. $A{A}^T{B}^T$
4. $B^T{A}^{T}A$

Q. More than One Answer Type
एक से अधिक उत्तर प्रकार के प्रश्न

If x = 3 + 2i and y = 3 – 2i, then

यदि x = 3 + 2i तथा y = 3 – 2i, तब

1. $\frac{x}{y}=\frac{5+12i}{13}$
2. x² + y² = 1
3. $\left|\frac{x}{y}\right|=1$
4. $|x|+|y|=2\sqrt{13}$

Q. If $A=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ then AA' =

1. $14$
2. $\left[\begin{array}{c}1\\ 3\\ 3\end{array}\right]$
3. $\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 3& 6& 9\end{array}\right]$
4. $Noneofthese$

Q. If P is $3\times 3$ matrix such that $P^T=2P+I,$, where $P^T$ is the transpose of P and I is the $3\times 3$ identity matrix, then there exists a column matrix $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\ne \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$ such that

1. $PX=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
2. PX = X
3. PX = 2X
4. PX = -X

Q. $\text{If}A+2B=\left[\begin{array}{cc}2& -4\\ 1& 6\end{array}\right],A^T+B^T=\left[\begin{array}{cc}1& 2\\ 0& -1\end{array}\right],\text{then}A=$

1. $\left[\begin{array}{cc}3& 0\\ 3& -8\end{array}\right]$
2. $\left[\begin{array}{cc}0& 4\\ -8& 3\end{array}\right]$
3. $\left[\begin{array}{cc}4& 0\\ 3& -8\end{array}\right]$
4. $\left[\begin{array}{cc}0& 4\\ 3& -8\end{array}\right]$

Q.

If $A=\left[\begin{array}{cccc}1& 0& 2& 3\\ 4& 2& 6& 4\end{array}\right]$ then fourth element of second column of $A^T$ =

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