Diagonal Matrix

Q. Construct a $2\times 2$ matrix, $A=\left[a_{ij}\right]$, whose elements are given by:
$\left(i\right)$ $a_{ij}=\frac{(i+j{)}^2}{2}$
$\left(ii\right)$ $a_{ij}=\frac{i}{j}$
$\left(iii\right)$ $a_{ij}=\frac{(i+2j{)}^2}{2}$

Q.

If $A$ is square matrix $A$ is its transpose, then $\frac{1}{2}(A–A)$ is

1. a symmetric matrix

2. a skew – symmetric matrix

3. a unit matrix

4. an elementary matrix

Q.

If $2i+4j-5k$ and $i+2j+3k$ are adjacent sides of a parallelogram, then the lengths of its diagonals are:

1. $7,\sqrt{69}$

2. $6,\sqrt{59}$

3. $5,\sqrt{65}$

4. $5,\sqrt{55}$

Q. If $A=$ dia. $(d_1,d_2,d_3,d_4)$ where $d_i>0\forall i=1,2,3,4$ is a diagonal matrix of order $4$ such that $d_1+2d_2+4d_3+8d_4=16,$ then the maximum value of $f\left(x\right)={\log }_{(\tan x+\cot x)}(det(A\left)\right)$ where $x\in (0,\frac{\pi }{2})$ is equal to

Q. If $A=\left[\begin{array}{ccc}1& 2& -3\\ 5& 0& 2\\ 1& -1& 1\end{array}\right],B=\left[\begin{array}{ccc}3& -1& 2\\ 4& 2& 5\\ 2& 0& 3\end{array}\right]$ and $C=\left[\begin{array}{ccc}4& 1& 2\\ 0& 3& 2\\ 1& -2& 3\end{array}\right],$ then compute $(A+B)$ and $(B-C).$ Also, verify that $A+(B-C)=(A+B)-C$.

Q. Assertion $\left(A\right)$:
The inverse of a matrix
$A=\left[\begin{array}{ccc}43& 1& 6\\ 35& 7& 4\\ 17& 3& 2\end{array}\right]$ does not exist.
Reason $\left(R\right)$:
The inverse of singular matrix is not possible

1. Both $\left(A\right)$ & $\left(R\right)$ are individually true & $\left(R\right)$ is correct explanation of $\left(A\right)$.
2. Both $\left(A\right)$ & $\left(R\right)$ are 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 matrix $A=\left[\begin{array}{cc}1& 0\\ 3& -1\end{array}\right],$ then $A^{15}$ can be

1. $A$
2. $A^2$
3. $A^3$
4. $A^4$

Q. Construct the matrix $A=[a_{ij}{]}_{3\times 3}$, where $a_{ij}=i-j$. Then $A$ is

1. Diagonal matrix
2. Skew-symmetric matrix
3. Symmetric matrix
4. Upper triangular matrix

Q. If $A=\left[\begin{array}{ccc}-1& 2& 3\\ 5& 7& 9\\ -2& 1& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}-4& 1& -5\\ 1& 2& 0\\ 1& 3& 1\end{array}\right]$, then verify that:
$\left(i\right)$ $(A+B{)}^{\prime }=A^{\prime }+B^{\prime }$
$\left(ii\right)$ $(A-B{)}^{\prime }=A^{\prime }-B^{\prime }$

Q. Let $A=\left[a_{ij}\right]$ is a diagonal matrix of order $4$ and $k$ is the possible number of zeros in $A.$ Then $k$ can be

1. $16$
2. $13$
3. $14$
4. $12$

Q. Find $\frac{1}{2}(A+A^{\prime })$ and $\frac{1}{2}(A-A^{\prime })$, when $A=\left[\begin{array}{ccc}0& a& b\\ -a& 0& c\\ -b& -c& 0\end{array}\right]$

Q. Consider a cuboid all of whose edges are integers and whose base is square. Suppose the sum of all its edges is numerically equal to the sum of the areas of all its six faces. Then the sum of all its edges is

1. $12$
2. $18$
3. $24$
4. $36$

Q. If $A$ is an orthogonal matrix of order $n$, then $|\text{adj}\left(\text{adj}\right(A\left)\right)|$ is

1. $1,$ when $n=4$
2. $-1,$ when $n=6$
3. $-1,$ when $n=7$
4. $1,$ when $n=5$

Q. Evaluate $\left|\begin{array}{cc}2& 4\\ -1& 2\end{array}\right|$

Q. Given $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}3& -1& 3\\ -1& 0& 2\end{array}\right],$ then find $2A-B$

Q. If $A=\left[\begin{array}{ccc}1& -3& 2\\ 2& 0& 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}2& -1& -1\\ 1& 0& -1\end{array}\right]$, then the matrix $C$ such that $A+B+C$ is a zero matrix, is

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

Q. Assertion :Let $A$ be a $2\times 2$ matrix with real entries. Let $I$ be the $2\times 2$ identity matrix. Denote by $tr\left(A\right)$, the sum of diagonal entries of $A$. Assume that $A^2=I$.
Reason: If $A\ne I$ and $A\ne -I$, then $tr\left(A\right)\ne 0$.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q. Identify the matrix given below:
$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 4& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -3\end{array}\right]$

1. diagonal matrix
2. unit matrix
3. scalar matrix
4. zero matrix

Q. Define a diagonal matrix.

Q. Let $P$ be a $2\times 2$ matrix such that $\left[\begin{array}{cc}1& 0\end{array}\right]P=-\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\end{array}\right]$ and $\left[\begin{array}{cc}0& 1\end{array}\right]P=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}-1& 1\end{array}\right]$
If $0$ and $I$ denote the zero and identity matrices of order $2$, respectively, then which of the following options is correct ?

1. $P^8-P^6+P^4+P^2=0$
2. $P^8+P^6-P^4+P^2=I$
3. $P^8+P^6+P^4-P^2=2I$
4. $P^8-P^6-P^4-P^2=0$

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 $A=\text{diagonal}(3,3,3)$, then $A^4=$

1. $27A$
2. $684A$
3. $81A$
4. $12A$

Q. Let $a$ and $b$ be two real numbers such that $a>1,b>1$. If $A=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$, then $\underset{n\to \infty }{lim}{A}^{-n}$ is

1. Unit matrix
2. Null matrix
3. $2I$
4. none of these

Q.

A diagonal matrix will always be

1. Rectangular matrix

2. Identity matrix

3. Square matrix

4. None of these

Q. Let $\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]$ The only correct statement about the matrix $A$ is

1. $A=(-1)I$
2. $A$ is a zero matrix
3. $A^{-1}$ does not exist
4. $A^2=I$

Q.

A main diagonal is a set of all elements which satisfy the condition i $\ne$ j.

1. True

2. False

Q. Construct a $2\times 3$ matrix $A=\left[a_{ij}\right]$ whose elements are given by $a_{ij}=2(i-j)$

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

Q. The value of $a_{22}-a_{13},$ in the matrix $A=\left[\begin{array}{ccc}1& 2& 0\\ -2& 3& 4\\ 3& 2& -1\end{array}\right],$ $($where $a_{ij}$ is the representation of element in the matrix$A)$ is

Q. If the shortest distance between the lines $\overrightarrow{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda \left(\begin{array}{c}2\hat{i}+3\hat{j}+4\hat{k}\end{array}\right)$ and $\overrightarrow{r}=2\hat{i}+4\hat{j}+5\hat{k}+\mu \left(\begin{array}{c}3\hat{i}+4\hat{j}+5\hat{k}\end{array}\right)$ is $k$, then the value of ${\tan }^{-1}\tan \left(\begin{array}{c}2\sqrt{6}k\end{array}\right)$ is

1. $1$
2. $2$
3. $2-\pi$
4. $\frac{\pi }{2}-2$

Q. For the matrix $A=\left[\begin{array}{cc}3& 2\\ 1& 1\end{array}\right],$ the values of ${}^{\prime }{a}^{\prime }$ and ${}^{\prime }{b}^{\prime }$ such that $A^2+aA+bI=0$ are

1. $2,3$
2. $1,4$
3. $-4,1$
4. $-2,3$