Transpose of a Matrix

Q.

The element of second row and third column in the inverse of $\begin{array}{c}\left[\begin{array}{ccc}1& 2& 1\\ 2& 1& 0\\ -1& 0& 1\end{array}\right]\end{array}$ is

1. -2

2. -1

3. 1

4. 2

Q.

Define symmetric and skew-symmetric matrix

Q.

Write a square matrix of order $2$, which is both symmetric and skew-symmetric.

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. None of these

2. XY

3. YX

4. -YX

Q.

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

1. Unique Solution

2. exactly two solutions

3. infinitely many solutions

4. No solution

Q.

Let M$\times \left[\begin{array}{cc}1& 1\\ 0& 2\end{array}\right]=\left[\begin{array}{cc}1& 2\end{array}\right]$, where M is a matrix.

a) State the order of the matrix M.

b) Find the matrix M

Q.

If A and B are symmetric matrices and AB = BA, then $A^{-1}B$ is a

1. Unit matrix

2. Symmetric matrix

3. Skew-symmetric matrix

4. Column Matrix

Q.

If $A^T$ and $B^T$ are transpose matrices of the square matrices$A$ and $B$ respectively then ${\left(AB\right)}^T$ is equal to

1. $A^T{B}^T$

2. $A{B}^T$

3. $B{A}^T$

4. $B^T{A}^T$

Q.

From the following, find the correct relation

1. (AB)' = A'B'

2. (AB)' = B'A'

3. AB = A'B'

4. (AB)' = BA

Q.

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

1. True

2. False

Q.

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

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

Q. Which of the following statements is true regarding the transpose of a matrix?

1. The transpose of a matrix will have same order as the matrix itself.
2. For any 2 matrices $A$ and $B$, if $AB$ exists that means $B^T{A}^T$ also exists.
3. For any 2 matrices $A$ and $B$, if $AB$ exists that means $A^T{B}^T$ also exists.
4. The multiplication of a matrix and it's transpose is not possible.

Q.

given : $M=\left[\begin{array}{cc}5& -3\\ -2& 4\end{array}\right]$
Find its transopose
Matrix M ' If possible, find:
(i) M + M ' (ii) M - M '

Q. If A is a square matrix $A+A^T$ is symmetric matrix, then
$A-A^T=$

1. Unit matrix
2. Symmetric matrix
3. Skew symmetric matrix
4. Zero matrix

Q. Let A and B are two non-singular square matrices, $A^T$ and $B^T$ are the transpose matrices of A and B respectively, then which of the following is correct

1. $B^{T}AB$ is symmetric matrix if and only if A is symmetric
2. $B^{T}AB$ is symmetric matrix if and only if B is symmetric
3. $B^{T}AB$ is skew symmetric matrix for every matrix A
4. $B^{T}AB$ is skew symmetric matrix if B is skew symmetric

Q.

Consider a matrix A. Then $(A^T{)}^T$ =​ ___

Q. $IfmatrixA=[a_{ij}{]}_{3\times 3},matrixB=[b_{ij}{]}_{3\times 3}where{a}_{ij}+a_{ji}=0and{b}_{ij}-b_{ji}=0,then{A}^4.B^{3}is$

1. skew-symmetric matrix
2. singular
3. symmetric
4. zero matrix

Q.

If $A^T=\left[\begin{array}{cc}3& 4\\ -1& 2\\ 0& 1\end{array}\right]\text{and}B=\left[\begin{array}{ccc}-1& 2& 1\\ 1& 2& 3\end{array}\right]$,

then find $A^T-B^T$

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

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

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

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

Q.

What is $A^T$ in Matrix?

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

1. XY
2. – YX
3. None of these
4. YX

Q. If $\left[\begin{array}{cc}a+2b& 2c-d\\ c+4d& 4b-a\end{array}\right]=\left[\begin{array}{cc}10& 2\\ 37& 14\end{array}\right]$, then find the values of $a,b,c,d$ respectively.

Q. If A is a $3\times 3$ non-singular matrix such that $A{A}^T=A^{T}AandB=A^{-1}{A}^T,thenB{B}^T$ is equal to

1. $B^{-1}$
2. ${\left(B^{-1}\right)}^T$
3. I
4. I + B

Q.

Carry out the multiplication of the expressions for the following pairs $\mathrm{ab},\mathrm{a}-\mathrm{b}$.

Q. If A is $3\times 4$ matrix and B is a matrix such that A'B and BA' are both defined. Then B is of the type

1. $3\times 4$
2. $3\times 3$
3. $4\times 4$
4. $4\times 3$

Q. If $3A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ x& 2& y\end{array}\right]$ and $A\cdot A^T=I$, then find the value of $x+y$. [3 Marks]

Q.

If A=$\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$ and I $=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, find 5A-3I

Q.

Transpose of $\left[\begin{array}{ccccc}1& 2& 3& 4& 5\end{array}\right]$ is a ___matrix

1. Column Matrix
   

2. Unit matrix

3. Square matrix

4. Row matrix

Q.

If $A=\left[\begin{array}{ccc}2& 8& -4\\ 0& -1& -2\\ 3& -1& 5\end{array}\right]$ Find A+A'

1. $\left[\begin{array}{ccc}4& 8& -1\\ 8& -2& -3\\ -1& -3& 10\end{array}\right]$

2. $\left[\begin{array}{ccc}4& 8& 1\\ 8& 2& 3\\ 1& 3& 10\end{array}\right]$

3. $\left[\begin{array}{ccc}4& 8& -1\\ 8& 6& -3\\ 1& 5& 10\end{array}\right]$

4. $\left[\begin{array}{ccc}4& 8& -1\\ 8& -2& -3\\ 0& -1& 9\end{array}\right]$

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

Q. The $n^{th}$ term of the series $4+14+30+52+80+114+\dots$ is

1. $2n^2+2n$
2. $3n^2+n$
3. $2n^2+2$
4. $5n-1$