Multiplication of Matrices

Q. Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of $A^2$ is $1,$ then the possible number of such matrices is :

1. $6$
2. $1$
3. $4$
4. $12$

Q.

Express $\cos \left(2x\right)$ in terms of $\tan \left(x\right)$.

Q.

1 nibble equals to how many bits and bytes ?

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. $3$
4. $6$

Q.

Let vector $a=i-2j+k$ and vector $b=i-j+k$ be two vectors. If vector $c$ is a vector such that $bxc=bxa$ and $c·a=0$, then $c·b$ is equal to?

1. $\frac{1}{2}$

2. $-1$

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

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

Q. If matrix $A={\left[a_{ij}\right]}_{2\times 2}$, where $a_{ij}=\left\{\begin{array}{ll}1& ,\mathrm{if}i\ne j\\ 0& ,\mathrm{if}i=j\end{array}\right.$, then A² is equal to

(a) I
(b) A
(c) O
(d) −I

Q. If $A=\left[\begin{array}{ccc}1& 1& -1\\ 2& 0& 3\\ 3& -1& 2\end{array}\right],B=\left[\begin{array}{cc}1& 3\\ 0& 2\\ -1& 4\end{array}\right]$ and $C=\left[\begin{array}{cccc}1& 2& 3& -4\\ 2& 0& -2& 1\end{array}\right]$, find $A\left(BC\right),\left(AB\right)C$ and show that $\left(AB\right)C=A\left(BC\right).$

Q. $\text{Let}P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]$ 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. $52$
2. $103$
3. $201$
4. $205$

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

Q.

What is the value of $2\sqrt{2}$?

Q.

If $x+\frac{1}{x}=6$ find $x-\frac{1}{x}$.

Q. Let M and N be two 3 $\times$ 3 matrices such that MN = NM. Further, if $M\ne N^{2}and{M}^2=N^4$, then

1. determinant of $(M^2+M{N}^2)$ is 0.
2. there is a 3 $\times$ 3 non - zero matrix U such that $(M^2+M{N}^2)$ U is zero matrix.
3. determinant of $(M^2+M{N}^2)\ge 1$
4. for a 3 $\times$ 3 matrix U, if $(M^2+M{N}^2)$Uequals the zero matrix, then U is the zero matrix

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.

Can you multiply a $3x2$ and $2x2$ matrix?

Q.

If $a$ and $b$ are real numbers such that ${(2+\alpha )}^4=a+b\alpha$, where $\alpha =\frac{(-1+i\sqrt{3})}{2}$ then $a+b$ is equal to:

1. $33$

2. $57$

3. $9$

4. $24$

Q.

If A is a square matrix of order 3 and |A| = 5 then |3A| = ?

1. 45

2. 32

3. 15

4. 135

Q. If $F\left(x\right)=\left[\begin{array}{ccc}\cos x& -\sin x& 0\\ \sin x& \cos x& 0\\ 0& 0& 1\end{array}\right],$ show that $F\left(x\right)F\left(y\right)=F(x+y)$.

Q. If P and Q are symmetric matrices of the same order then PQ-QP is

1. Identity matrix
2. Zero matrix
3. Skew symmetric matrix
4. Symmetric matrix

Q. Determine the product
$\left[\begin{array}{ccc}-4& 4& 4\\ -7& 1& 3\\ 5& -3& -1\end{array}\right]\left[\begin{array}{ccc}1& -1& 1\\ 1& -2& -2\\ 2& 1& 3\end{array}\right]$
and use it to solve the system of equations
$x-y+z=4,x-2y-2z=9,2x+y+3z=1$.

Q. Let $A$ and $B$ be two square matrices of order $3$ such that $AB=A$ and $BA=B.$ If $(A+B{)}^{10}=k(A+B),$ then the value of $k$ is

Q. The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

Q.

If $AandB$ are square matrices of equal degree, then which one is correct among the following?

1. $A+B=B+A$

2. $A+B=A–B$

3. $A–B=B–A$

4. $AB=BA$

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

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

Q.

If $a^{1/x}=b^{1/y}=c^{1/z}$and $b^2=ac$, then $x+z=$

1. $y$

2. $2y$

3. $2xyz$

4. None of these

Q. Let $A$ be a $3\times 3$ matrix having entries from the set $\{–1,0,1\}$. The number of all such matrices $A$ having sum of all the entries equal to $5$, is .

Q.

Prove that cos5A= 16cos⁵A -20cos³A +5cosA

Q. If $P=\left[\begin{array}{cc}\sqrt{3}/2& 1/2\\ -1/2& \sqrt{3}/2\end{array}\right],A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\text{and}Q=PA{P}^T,\text{then}{P}^T{Q}^{2005}P\text{is}$

1. $\left[\begin{array}{cc}1& 2005\\ 0& 1\end{array}\right]$
2. $\left[\begin{array}{cc}1& 2005\\ 2005& 1\end{array}\right]$
3. $\left[\begin{array}{cc}1& 0\\ 2005& 1\end{array}\right]$
4. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Q.

if A is any square matrix of order 3x3 then |3A| is

Q.

Let $z_1,z_2$ be two complex numbers such that $z_1+z_{2}and{z}_1{z}_2$ both are real, then

1. $z_1=-z_2$

2. $z_1=\overline{z_2}$

3. $z_1=-\overline{z_2}$

4. $z_1=z_2$

Q.

If $e^y+xy=e,$ then the value of $\frac{d^{2}y}{d{x}^2}forx=0$is

1. $\frac{1}{e}$

2. $\frac{1}{e^2}$

3. $\frac{1}{e^3}$

4. none of these