Inverse of a matrix

Q.

$IfA=\left[\begin{array}{ccc}cosx& sinx& 0\\ -sinx& cosx& 0\\ 0& 0& 1\end{array}\right]=f\left(x\right),then{A}^{-1}$ is equal to

1. -f(x)

2. -f(-x)

3. f(-x)

4. f(x)

Q.

What is adjoint and inverse of a matrix?

Q. If P, Q and R are $n\times n$ matrix and
det(P) = 2, det(Q) = 3 and det(R) = 5,
then find the value of $\det \left(P^2{\mathrm{QR}}^{-1}\right).$

1. $\frac{5}{7}$
2. $\frac{12}{5}$
3. $\frac{5}{12}$
4. $\frac{7}{5}$

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

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

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

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

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. 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.

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