Inverse of a Matrix

Q. If $A^3=0$, then $I+A+A^2$ equals

1. $I-A$
2. $(I-A{)}^{-1}$
3. $(I+A{)}^{-1}$
4. $noneofthese$

Q. The inverse of $\left[\begin{array}{ccc}3& 5& 7\\ 2& -3& 1\\ 1& 1& 2\end{array}\right]$ is

1. $\left[\begin{array}{ccc}7& 3& -26\\ 3& 1& -11\\ -5& -2& 0\end{array}\right]$
2. $\left[\begin{array}{ccc}7& 3& -26\\ 3& 1& 11\\ -5& -2& 1\end{array}\right]$
3. $\left[\begin{array}{ccc}3& 1& 11\\ 7& 3& -26\\ -5& 2& 1\end{array}\right]$
4. $Noneofthese$

Q. If $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]A\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then $A$ is equal to

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

Q. If $a,b,c$ and $d$ are real numbers such that $a^2+b^2+c^2+d^2=1$ and if $A=\left[\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right],$where $i^2=-1,$ then $A^{-1}=$

Q. If matrix $A=\left[\begin{array}{ccc}1& 0& -1\\ 3& 4& 5\\ 0& 6& 7\end{array}\right]$ and its inverse is denoted by $A^{-1}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$, then the value of $a_{23}$=

1. $\frac{21}{20}$
2. $\frac{1}{5}$
3. $-\frac{2}{5}$
4. $\frac{2}{5}$

Q. What is the inverse of the matrix $\left[\begin{array}{cc}-3& 2\\ 5& -1\end{array}\right]$

1. $\left[\begin{array}{cc}3& 5\\ 2& 1\end{array}\right]$
2. $\frac{1}{7}\left[\begin{array}{cc}3& 5\\ 2& 1\end{array}\right]$
3. $\frac{1}{7}\left[\begin{array}{cc}-1& -2\\ -5& -3\end{array}\right]$
4. $\frac{1}{7}\left[\begin{array}{cc}1& 2\\ 5& 3\end{array}\right]$

Q.

Find the inverse of the matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 1& 1& 1\\ 2& 3& 4\end{array}\right]$

1. $\left[\begin{array}{ccc}1& 5& 2\\ 3& 1& -2\\ 1& 2& 3\end{array}\right]$

2. $\left[\begin{array}{ccc}1& 2& 3\\ 1& 5& 9\\ 3& 7& 2\end{array}\right]$

3. $\left[\begin{array}{ccc}7& 9& 1\\ 3& 5& 7\\ 2& 1& 3\end{array}\right]$

4. Inverse doesn’t exist

Q. Let $P$ be a non-singular matrix such that $I+P+P^2+\cdots +P^n=O$. Then $P^{-1}$ is equal to

1. $P$
2. $-P^n$
3. $P^{n-1}$
4. $P^n$

Q. If $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]A\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$ then the sum of all the elements of matrix $A$ is

Q. Let $A=\left[a_{ij}\right]$ be $3\times 3$ matrix given by
$a_{ij}=\left\{\begin{array}{cc}\left(\frac{i+j}{2}\right)+\frac{|i-j|}{2},& ifi\ne j\\ \frac{i^j-(i.j)}{i^2+j^2},& ifi=j\end{array}\right\}$
where $a_{ij}$ denotes element of $i^{th}$ row & $j^{th}$ column of matrix A.If $A=p{A}^2-q{A}^{-1}-rI$, then remainder when $(p+q+r{)}^{11}$ is divided by 7 is

Q. If $A$ and $B$ are two square matrices such that $B=-A^{-1}BA,$ then $(A+B{)}^2$ is equal to

Q. Let M be a $2\times 2$ symmetric matrix with integer entries. Then, M is invertible if

1. the first column of M is the transpose of the second row of M
2. the second row of M is the transpose of the first column of M
3. M is a diagonal matrix with non-zero entries in the main diagonal
4. the product of entries in the main diagonal of M is not the square of an integer

Q. If $\Delta =\left|\begin{array}{ccc}abc& b^{2}c& c^{2}b\\ abc& c^{2}a& c{a}^2\\ abc& a^{2}b& b^{2}a\end{array}\right|=0$ $(a,b,c\in R$
and are all different and non-zero), then the value of $a+b+c$ is:

1. $3$
2. $-1$
3. $1$
4. $0$

Q. If A, B and C are 3 invertible matrices and BAC = I, what is the value of A.

1. CB
2. $C^{-1}{B}^{-1}$
3. $B^{-1}{C}^{-1}$
4. $A^{-1}$

Q. If $A$ and $B$ are square matrices of same order satisfying $A+B=AB,$ then

1. $AB=BA$
2. $AB=-BA$
3. $A+B=-BA$
4. $A-B=BA$

Q. If $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]A\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then $A=$

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

Q. If A and B are two square matrices such that $B=-A^{-1}$ BA, then $(A+B{)}^2=$

1. $0$
2. $A^2+B^2$
3. $A^2+2AB+B^2$
4. $A+B$

Q. If $A$ and $B$ are non-singular, symmetric and commutable matrices, then $A^{-1}{B}^{-1}$ is

1. Symmetric matrix
2. Identity matrix
3. Skew-symmetric matrix
4. None of these

Q. Let $F\left(\alpha \right)=\left[\begin{array}{ccc}cos\alpha & -sin\alpha & 0\\ sin\alpha & cos\alpha & 0\\ 0& 0& 1\end{array}\right]$, where $\alpha ϵR.$
Then $\left[F\right(\alpha ){]}^{-1}$ is equal to

1. $F(-\alpha )$
2. $F(-{\alpha }^{-1})$
3. $F\left(2\alpha \right)$
4. $Noneofthese$

Q. The inverse of the matrix $\left[\begin{array}{ccc}7& -3& -3\\ -1& 1& 0\\ -1& 0& 1\end{array}\right]$ is

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

Q. If $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]A\left[\begin{array}{cc}-3& 2\\ 5& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then A =

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

Q. A and B are square matrices and A is non-singular matrix, $(A^{-1}BA{)}^n,nϵI^{+}$ is equal to

1. $A^{-n}{B}^n{A}^n$
2. $A^n{B}^n{A}^{-n}$
3. $A^{-1}{B}^{n}A$
4. $A^{-1}B{A}^n$

Q. If $M=\left[\begin{array}{cc}1& -3\\ -1& 1\end{array}\right],$ then the value of $M-\frac{1}{3}{M}^2+\frac{1}{9}{M}^3-\frac{1}{27}{M}^4+....\infty$ is

1. $\frac{1}{13}\left[\begin{array}{cc}-1& 9\\ 3& -1\end{array}\right]$
2. $\frac{3}{13}\left[\begin{array}{cc}-1& 9\\ 3& -1\end{array}\right]$
3. $\frac{1}{13}\left[\begin{array}{cc}1& -9\\ -3& 1\end{array}\right]$
4. $\frac{3}{13}\left[\begin{array}{cc}1& -9\\ -3& 1\end{array}\right]$

Q. A square matrix A is said to be nilpotent of index m. $If{A}^m=0,now,ifforthisA,(I-A{)}^n=I+A+A^2+...+A^{m-1},thennisequalto$

1. \N
2. m
3. - m
4. - 1

Q. If $A$ and $B$ are square matrices of the same order and $A$ is non-singular, then for a positive integer $n,(A^{-1}BA{)}^n$ is equal to

1. $A^{-n}{B}^n{A}^n$
2. $A^n{B}^n{A}^{-n}$
3. $A^{-1}{B}^{n}A$
4. $n\left(A^{-1}BA\right)$

Q. Let $A=diag(x^2,6x,1)$ and $B=diag(\frac{1}{x^2},\frac{1}{x},1)$ be two diagonal matrices. A function $f$ is defined as $f\left(x\right)=tr\left(\right(A^{-1}B{)}^{-1}).$ If $\int \frac{f\left(x\right)}{x^2+1}dx=g\left(x\right)$ where $g\left(0\right)=0,$ then the number of roots of $g\left(x\right)=0$ is

$\left[tr\right(P)$ denotes the trace of matrix $P]$

Q. If $B=\left[\begin{array}{ccc}5& 2\alpha & 1\\ 0& 2& 1\\ \alpha & 3& -1\end{array}\right]$ is the inverse of a $3\times 3$ matrix $A,$ then the sum of all values of $\alpha$ for which $det\left(A\right)+I=0,$ is

1. $2$
2. $1$
3. $0$
4. $-1$

Q. If $A=\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& a& 1\end{array}\right]\text{and}{A}^{-1}=\left[\begin{array}{ccc}1/2& -1/2& 1/2\\ -4& 3& c\\ 5/2& -3/2& 1/2\end{array}\right],$ then $a+c=$