Inverse of a Matrix

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 $A=\left[\begin{array}{cc}1& tan\frac{\theta }{2}\\ -tan\frac{\theta }{2}& 1\end{array}\right]$ and AB = I, then B =

1. $\text{None of these}$
2. $co{s}^2\frac{\theta }{2}.A$
3. $co{s}^2\frac{\theta }{2}.I$
4. $co{s}^2\frac{\theta }{2}.A^T$

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

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

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

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

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. 0
2. m
3. - m
4. - 1

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 A, B, C are three square matrices such that AB = AC implies B = C, then the matrix A is always a/an

1. Diagonal matrix
2. Singular matrix
3. Non-singular matrix
4. Orthogonal matrix

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. Let A, B be square matrix such that A B = 0 and B is non singular then

1. $\left|A\right|$ must be zero but A may non zero
2. nothing can be said in general about A
3. none of these
4. A must be zero matrix

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

1. $f(\alpha -1)$
2. $f(-\alpha )$
3. $f\left(\alpha \right)$
4. $none$