Inverse of a Matrix

Q.

Find the value of ' a ' and ' b ' if 5+2√3 / 7+4√3 = a+b√3

Q.

The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2\times 2$ matrix such that the trace of $A$ is $3$ and trace of $A^3$ is $-18$, then the value of the determinant of $A$ is ___________

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

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

Q.

Prove $\tan \left(3\theta \right)=\frac{(3\tan \theta -{\tan }^3\theta )}{(1-3{\tan }^2\theta )}$

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

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 A and B are any two different square matrices of order n with A – B is non-singular $A^3=B^3$ and A(AB) = B(BA) , then

1. $A^2+B^2=O$
2. $A^2+B^2=I$
3. $A^2+B^3=I$
4. $A^3+B^3=O$

Q. If $A$ and $B$ are non - zero square matrices of the same order such that $AB=0$, then

1. adj $A=0$ and adj $B=0$
2. $|A|=0$ or $|B|=0$
3. adj $A=0$ or adj $B=0$
4. None of these

Q. Consider the following statements:
1. If $A$ and $B$ are two square matrices of same order, then $(A+B)(A-B)=A^2-B^2$.
2. If $A$ and $B$ are two square matrices of same order, then $(AB{)}^n=A^n{B}^n$.
3. If $A$ and $B$ are two matrices such that $AB=A$ and $BA=B$, then $A$ and $B$ are idempotent.
Which of these is/are not correct?

1. $\left(2\right)and\left(3\right)$
2. $\left(1\right)and\left(2\right)$
3. All of these
4. $\left(3\right)and\left(1\right)$

Q. Let $P=\left[\begin{array}{c}1\\ 4\\ 16\end{array}\begin{array}{c}0\\ 1\\ 4\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}\right]$ and $I$ be the identity matrix of order $3$. If $Q=\left[q_{ij}\right]$ is a matrix such that $P^{50}-Q=I$, then $\frac{q_{31}+q_{32}}{q_{21}}$ equals

1. 103
2. 205
3. 201
4. 52

Q. For integral value of m if the value of $\frac{2}{m}+3m=7$, find the value of $9m^2-\frac{4}{m^2}$ using identities.

1. 77
2. 63
3. 42
4. 35

Q. Let $A,B,C,D$ be (not necessarily square) real matrices such that $A^T=BCD;B^T=CDA;C^T=DAB$ and $D^T=ABC$ for the matrix $S=ABCD$, then which of the following is/are true

1. $S^3=S$
2. $S^2=S^4$
3. $S=S^2$
4. none of these

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

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

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. A must be zero matrix
3. nothing can be said in general about A
4. none of these

Q. The transpose of a square matrix is a

1. rectangular matrix
2. diagonal matrix
3. square matrix
4. scaler matrix

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

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

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. Inverse doesn’t exist

2. 

3. 

4. 

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. If A and B are matrices of the same order, then ${(A+B)}^2=A^2+2AB+B^2$ is possible, iff

1. AB= I
2. BA= I
3. AB= BA
4. none of these

Q.

Which of the following is the inverse pair of 1 under the operations multiplication and subtraction?

1. 1, 1

2. 1, 0

3. 0, 0

4. None of these

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. Without assuming the formula, find the sum of the series $1^7+2^7+3^7+....+n^7$.

Q.

$\frac{\tan \theta }{1-cot\theta }+\frac{cot\theta }{1-\tan \theta }=1+\tan \theta +cot\theta$.

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. 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. Grafical representation of a quaratic polynomial

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 A is a $3\times 3$ non-singular matrix such that $A{A}^T=A^{T}AandB=A^{-1}{A}^T,thenB{B}^T$ is equal to

1. I
2. I + B
3. $B^{-1}$
4. ${\left(B^{-1}\right)}^T$