Transpose of a Matrix

Q.

If $P$ is a $3\times 3$matrix such that ${\mathrm{P}}^{\mathrm{T}}=2\mathrm{P}+\mathrm{I}$ , where ${\mathrm{P}}^{\mathrm{T}}$ 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. $\mathrm{PX}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

2. $\mathrm{PX}$

3. $\mathrm{PX}=2\mathrm{X}$

4. $\mathrm{PX}=-\mathrm{X}$

Q.

What is a row matrix?

Q. Prove that a² + b² + c² −ab−bc−ca is always non-negative for all values of a, b, and c.

Q.

If A and B are symmetric matrices of the same order and X = AB + BA and Y = AB - BA, then $(XY{)}^T$ is equal to

1. XY

2. YX

3. -YX

4. None of these

Q. If A, B, C are nonsingular square matrices of order $3\times 3$ then which of the following is not necessarily true?

1. $A^{T}A=I\Rightarrow A{A}^T=I$
2. $(A-B)C=AC-BC$
3. $I^{-1}=1$
4. $A^{-1}{B}^{-1}=(BA{)}^{-1}$

Q. Find the remainder when $7^{21}+7^{22}+7^{23}+7^{24}$ is divided by $25$

1. $0$
2. $2$
3. $4$
4. $6$

Q. If A', B' are transpose matrices of the square matrices A, B respectively , then (AB)' is equal to

1. AB'
2. A'B'
3. B'A'
4. BA'

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

Q. $A=[a_{ij}{]}_{n\times n}$ and $a_{ij}=i^2-j^2$ then A is necessarily

1. skew symmetric matrix
2. zero matrix
3. a unit matrix
4. symmetric matrix

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. Let $M$ be a $3\times 3$ non-singular matrix with $det\left(M\right)=\alpha$. If $\left[M^{-1}adj\right(adj\left(M\right)]=KI$, then the value of $K$ is

1. $1$
2. ${\alpha }^2$
3. $\alpha$
4. ${\alpha }^3$

Q. Let M and N be two 3 $\times$ 3 skew - symmetric matrices such that MN = NM. If $P^T$ denotes the transpose of P, then $M^2{N}^2\left(M^{T}N{)}^{-1}\right(M{N}^{-1}{)}^T$ is equal to

1. $M^2$
2. $-N^2$
3. $-M^2$
4. $MN$

Q.

To represent 2.5162 which of the following method would be used.

1. By using square identities

2. By the process of successive magnification

3. By using Pythagoras theorem (Geometrical construction)

4. By constructing equal arcs

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 $A^T$ =

___

Q. Evaluate : $(b-7{)}^2$

Q.

If A and B are two square matrices which are skew symmetric then$(AB{)}^T$ equals to

1. AB
2. BA
3. -AB
4. -BA

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. $Noneofthese$