Adjoint of a Matrix

Q. If $\text{adj}B=A,|P|=|Q|=1,$ then $\text{adj}\left(Q^{-1}B{P}^{-1}\right)$ is

1. $PQ$
2. $QAP$
3. $PAQ$
4. $P{A}^{-1}Q$

Q. If the square ABCD where A (0, 0), B(2, 0), C(2, 2) and D(0, 2) undergoes the following transformations successively
(i) $f_1$(x, y) $\to$ (y, x).
(ii) $f_2$(x, y) $\to$ (x + 3y, y).
(iii) $f_3$(x, y) $\to$$(\frac{x-y}{2},\frac{x+y}{2})$
then the final figure is a

1. Square
2. Parallelogram
3. rhombus
4. 2x - y = a

Q. Match the properties of adjoint of a matrix :-

1. ${\left(AdjA\right)}^T$
2. $AdjB.AdjA$
3. ${\left(A\right)}^{{}^{N-1}}$ , N is order of $\left(A\right)$

Q. Let $A$ be a $3\times 3$ square matrix. If $B=adj\left(A\right),$ $C=adj\left(adj\right(A\left)\right)$ and $D=adj\left(adj\right(adj\left(A\right)\left)\right),$ then $|adj\left(adj\right(adj\left(adj\right(ABCD\left)\right)\left)\right)|,$ in terms of $|A|$ is

1. $|A^{60}|$
2. $|A^{120}|$
3. $|A^{180}|$
4. $|A^{240}|$

Q. The number of different matrices can be formed using all the letters of word $TOMATO,$ is

1. $720$
2. $900$
3. $180$
4. $360$

Q. If $A=\left[\begin{array}{ccc}0& c& -b\\ -c& 0& a\\ b& -a& 0\end{array}\right]$ and $B=\left[\begin{array}{ccc}{a}^2& ab& ac\\ ab& b^2& bc\\ ac& bc& c^2\end{array}\right]$, then $AB=$

1. $B$
2. $I$
3. $O$
4. $A$

Q. which of the following is true, if A is an invertible square matrix :-

1. ${\left(A^T\right)}^{-1}={\left(A^{-1}\right)}^T$
2. ${\left(A^T\right)}^{-1}=\left(A^{-1}\right)\left(A^T\right)$
3. $\left(A^T\right)={\left(A^{-1}\right)}^T$
4. ${\left(A^T\right)}^{-1}=\left(A^T\right)\left(A^{-1}\right)$

Q. The value of $\left|\begin{array}{cc}2+i& 2-i\\ 1+i& 1-i\end{array}\right|$ is, where $(i^2=-1)$

1. $1-i$
2. purely imaginary
3. $1+i$
4. a real quantity

Q. Let $P=\left[a_{ij}\right]$ be a $3\times 3$ invertible matrix, where $a_{ij}\in \{0,1\}$ for $1\le i,j\le 3$ and exactly four elements of $P$ are $1$. If $N$ denotes the number of such possible matrices $P$, then which of the following is/are true?

1. Number of divisors of $N$ is even.
2. Sum of divisors of $N$ is $91$.
3. Determinant of $\text{adj}\left(P\right)$ can be $-1$.
4. Determinant of $\text{adj}\left(P\right)$ can be $1$.

Q. If $f\left(x\right)=|x-1|-\left[x\right]$, where $\left[x\right]$= the greatest integer less than or equal to $x$, then

1. $f(1+0)=1,f(1-0)=0$
2. $f(1+0)=-1,0=f(1-0)$
3. ${lim}_{x\to 1}f\left(x\right)$ exist
4. ${lim}_{x\to 1}f\left(x\right)$ does not exist

Q. If A is a diagonal matrix of order 3 $\times$ 3 is commutative with every square matrix of order 3 $\times$ 3 under multiplication and trace (A) = 12, then

1. $\left|A\right|=64$
2. $\left|A\right|=16$
3. $\left|A\right|=12$
4. $\left|A\right|=0$

Q. Let $A=[a_{ij}{]}_{4\times 4}$ be a matrix such that $a_{ij}=\{\begin{array}{cc}2,& \text{if}i=j\\ 0,& \text{if}i\ne j\end{array}.$
Then the value of $\left\{\frac{det(adj\left(adjA\right))}{7}\right\}$ is
$\left(\right\{.\}$ represents the fractional part function $)$

1. $\frac{1}{7}$
2. $\frac{2}{7}$
3. $\frac{3}{7}$
4. $\frac{6}{7}$

Q. If A is $6\times 6$ matrix and $||A|adj\left(|A|A\right)|=|A{|}^n$, then $n$ is

1. 40
2. 31
3. 25
4. 41

Q. If $\alpha$ is a characteristic root of a non-singular matrix $A$, then characteristic root of $adjA$ is

Q. If $A=\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right],\text{then}adj(3A^2+12A)$ is equal to

1. $\left[\begin{array}{cc}72& -84\\ -63& 51\end{array}\right]$
2. $\left[\begin{array}{cc}51& 63\\ 84& 72\end{array}\right]$
3. $\left[\begin{array}{cc}51& 84\\ 63& 72\end{array}\right]$
4. $\left[\begin{array}{cc}72& -63\\ -84& 51\end{array}\right]$