Multiplication of Matrices

Q. If for the matrix, $A=\left[\begin{array}{cc}1& -\alpha \\ \alpha & \beta \end{array}\right],A{A}^T=I_2,$ then the value of ${\alpha }^4+{\beta }^4$ is:

1. $1$
2. $3$
3. $2$
4. $4$

Q. Let $A=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right],(\alpha \in R)$ such that $A^{32}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$. Then a value of $\alpha$ is:

1. $0$
2. $\frac{\pi }{64}$
3. $\frac{\pi }{32}$
4. $\frac{\pi }{16}$

Q.

Let $A$ be a $2\times 2$real matrix with entries from $\left\{0,1\right\}$ and $A\ne 0$. Consider the following two statements:

(P) If $A\ne I_2$then $\left|A\right|=-1$

(Q) If $\left|A\right|=1$ , then $tr\left(A\right)=2$,

where ${\text{I}}_2$ denotes $2\times 2$ identity matrix and $tr\left(A\right)$ denotes the sum of the diagonal entries of $A$. Then:

1. Both (P) and (Q) are false

2. (P) is true and (Q) is false

3. Both (P) and (Q) are true

4. (P) is false and (Q) is true

Q. Let $A$ and $B$ be two $2\times 2$ matrices. Consider the statements
$\left(i\right)AB=0\Rightarrow A=0$ or $B=0$
$\left(ii\right)AB=I_2\Rightarrow A=B^{-1}$
$\left(iii\right)(A+B{)}^2=A^2+2AB+B^2$

1. (i) and (ii) are false, (iii) is true
2. (ii) and (iii) are false, (i) is true
3. (i) is false, (ii) and (iii) is true
4. (i) and (iii) are false, (ii) is true

Q. Let $A=\left[\begin{array}{ccc}2& 0& 7\\ 0& 1& 0\\ 1& -2& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}-x& 14x& 7x\\ 0& 1& 0\\ x& -4x& -2x\end{array}\right]$ be two matrices such that $AB=(AB{)}^{-1}$ and $AB\ne I,$ where $I$ is an identity matrix of order $3\times 3.$ Then the value of $tr(AB+(AB{)}^2+(AB{)}^3+\cdots +(AB{)}^{100})$ is
$($ Here, $tr\left(A\right)$ denotes the trace of matrix $A,$ i.e., sum of diagonal elements of $A.)$

1. $150$
2. $300$
3. $200$
4. $100$

Q. Let,
$A=\left[\begin{array}{ccc}4& 6& -1\\ 3& 0& 2\\ 1& -2& 5\end{array}\right],B=\left[\begin{array}{cc}2& 4\\ 0& 1\\ -1& 2\end{array}\right]$
$\text{and}C=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$
The expression which is not defined is:

1. $B^{\prime }B$
2. $CAB$
3. $A+B^{\prime }$
4. $A^2+A$

Q.

What is a full rank matrix?

Q.

Find the square of $71$ without multiplication.

Q. $IfA=\left[\begin{array}{cc}i& -i\\ -i& i\end{array}\right]andB=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right],then{A}^{8}equals$

1. 4B
2. 128 B
3. - 128 B
4. - 64 B

Q. A square matrix $A$ is said to be orthogonal if $A^{\prime }A=A{A}^{\prime }=I_n,A^{\prime }$ is transpose of $A.$
If $A$ and $B$ are orthogonal matrices, of the same order, then which one of the following is an orthogonal matrix

Q. If $A=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ and $A^2=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha \end{array}\right]$, then

1. $\alpha =a^2+b^2,\beta =ab$
2. $\alpha =a^2+b^2,\beta =2ab$
3. $\alpha =a^2+b^2,\beta =a^2-b^2$
4. $\alpha =2ab,\beta =a^2+b^2$

Q. Let $A=\left[\begin{array}{cc}i& -i\\ -i& i\end{array}\right],i=\sqrt{-1}$ then, the system of linear equations $A^8\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}8\\ 64\end{array}\right]$ has:

1. Infinitely many solutions
2. A unique solution
3. Exactly two solutions
4. No solution

Q. $$LetM=\left[\begin{array}{cc}{\sin }^4\theta & -1-{\sin }^2\theta \\ 1+{\cos }^2\theta & {\cos }^4\theta \end{array}\right]=\alpha I+\beta M^{-1},$$
Where $\alpha =\alpha \left(\theta \right)and\beta =\beta \left(\theta \right)$ are real numbers, and $I$ is the $2\times 2$ identity matrix. If ${\alpha }^{\ast }$ is the minimum of set $\left\{\alpha \right(\theta ):\theta \in [0,2\pi \left)\right\}$ and
${\beta }^{\ast }$ is the minimum of set $\left\{\beta \right(\theta ):\theta \in [0,2\pi \left]\right\}$, then the value of ${\alpha }^{\ast }$+${\beta }^{\ast }$ is

1. $-\frac{37}{16}$
2. $-\frac{31}{16}$
3. $-\frac{29}{16}$
4. $-\frac{17}{16}$

Q. Let A and B be square matrices of same order satisfying AB = A and BA = B. Then $A^2{B}^2$ equals, (O being the zero matrix of the same order as B)

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

Q. Let the matrix $M=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ satisfy $M^n=M^{n-2}+M^2-I$ for $n=3,4,5,6,\dots$ where $I$ is the identity matrix of order $3$.
Also $U_1,U_2,U_3$ are column matrices satisfying $M^{50}{U}_1=\left[\begin{array}{c}1\\ 25\\ 25\end{array}\right],M^{50}{U}_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],M^{50}{U}_3=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$ and $U$ is a $3\times 3$ matrix whose columns are $U_1,U_2,U_3$.
Then

1. $M^{50}=25M^2-24I$
2. $det\left(adj{M}^{50}\right)$ is equal to $1$
3. Let $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}0\\ 2\\ 4\end{array}\right]$ be two matrices. Then the system of equations $UX=B$ has infinitely many solutions.
4. Let $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}0\\ 2\\ 4\end{array}\right]$ be two matrices. Then the system of equations $UX=B$ has a unique solution.

Q. Let $a$ and $b$ be two real numbers such that $a>1,b>1$. If $A=\left(\begin{array}{cc}a& 0\\ 0& b\end{array}\right)$, then $\underset{n\to \infty }{lim}(A^{-1}{)}^n$ is

1. unit matrix
2. null matrix
3. $2I$, where $I$ is the identity matrix.
4. None of these

Q. The total number of matrices
$A=\left[\begin{array}{ccc}0& 2y& 1\\ 2x& y& -1\\ 2x& -y& 1\end{array}\right],(x,y\in R,x\ne y)$

for which $A^{T}A=3I_3$ is :

1. $2$
2. $4$
3. $6$
4. $3$

Q. Let $P=\left[\begin{array}{ccc}1& 0& 0\\ 3& 1& 0\\ 9& 3& 1\end{array}\right]$ and $Q=\left[q_{ij}\right]$ be two $3\times 3$ matrices such that $Q-P^5=I_3.$ Then $\frac{q_{21}+q_{31}}{q_{32}}$ is equal to

1. $9$
2. $10$
3. $15$
4. $135$

Q. For two matrices A and B of same order, if AB = A and BA = B then $A^2$is equal to?

1. I (identity matrix)
2. A
3. B
4. $\varphi \left(nullmatix\right)$

Q. Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ satisfies $A^n=A^{n-2}+A^2-I$ for $n\ge 3.$ And trace of a square matrix $X$ is equal to the sum of elements in its principal diagonal. Further consider a matrix ${\cup }_{3\times 3}$ with its column as ${\cup }_1,{\cup }_2,{\cup }_3$ such that $A^{50}{\cup }_1=\left[\begin{array}{c}1\\ 25\\ 25\end{array}\right],A^{50}{\cup }_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],A^{50}{\cup }_3=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$ Then,

The value of $|\cup |$ equals

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

Q. $\mathrm{When}\mathrm{two}\mathrm{non}-\mathrm{zero}\mathrm{matrices}\mathrm{are}\mathrm{multiplied},\mathrm{they}\mathrm{may}\mathrm{give}a\mathrm{zero}\mathrm{matrix}.$

1. False
2. True

Q. If A and B are two square matrices of order 3 $\times$ 3 which satisfy AB = A and BA = B then $(A+B{)}^7$ is

1. $7(A+B)$
2. ${7.1}_{3\times 3}$
3. $64(A+B)$
4. ${1281}_{3\times 3}$

Q.

$IfA=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right],thenfornϵN,A^n=$

1. A, if n is odd

2. A, if n is even

3. I, if n is odd

4. I, if n is even

Q. Suppose a Matrix A satisfies $A^2-5A+7I=0$ If $A^5=aA+bI$, then the values of 2a + b is.

1. -87
2. -105
3. 1453
4. 1155

Q.

Let $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $B=A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ are two matrices such that AB = BA and $c\ne 0$, then value of $\frac{a-d}{3b-c}$ is :

1. 0

2. 2

3. -2

4. -1

Q.

$IfF\left(x\right)=\left[\begin{array}{ccc}cosx& -sinx& 0\\ sinx& cosx& 0\\ 0& 0& 1\end{array}\right],thenF\left(x\right).F\left(y\right)=$

1. $F\left(xy\right)$

2. $F\left(x\right)+F\left(y\right)$

3. $F(x+y)$

4. $F(x-y)$

Q. If A is $3\times 4$ matrix and B is a matrix such that A'B and BA' are both defined. Then B is of the type

1. $3\times 4$
2. $3\times 3$
3. $4\times 4$
4. $4\times 3$

Q. Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ satisfies $A^n=A^{n-2}+A^2-I$ for $n\ge 3.$ And trace of a square matrix $X$ is equal to the sum of elements in its principal diagonal. Further consider a matrix ${\cup }_{3\times 3}$ with its column as ${\cup }_1,{\cup }_2,{\cup }_3$ such that $A^{50}{\cup }_1=\left[\begin{array}{c}1\\ 25\\ 25\end{array}\right],A^{50}{\cup }_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],A^{50}{\cup }_3=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$ Then,

The value of $|A^{50}|$ equals

1. $0$
2. $1$
3. $-1$
4. $25$

Q. Suppose a Matrix A satisfies $A^2-5A+7I=0$ If $A^5=aA+bI$, then the values of 2a + b is.

1. -87
2. -105
3. 1453
4. 1155

Q. If $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$, then the matrix $A^{-50}$ when $\theta =\frac{\pi }{12}$, is equal to :

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