Addition and Subtraction of a Matrix

Q.

Is ${\tan }^{2}x=se{c}^{2}x-1$ an identity?

Q.

$sec\left(\cos e{c}^{-1}x\right)$ is equal to

1. $\cos ec\left(se{c}^{-1}x\right)$

2. $cotx$

3. $\mathrm{\pi }$

4. None of these

Q. Two farmers Ramkrishan and Gurucharan Singh cultivates only three varieties of rice namely Basmati, Permal and Naura. The sale (in rupees) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices A and B.
September Sales (in Rupees)
$\begin{array}{ccccc}& & \text{Basmati}& \text{Permal}& \text{Naura}\end{array}$
$A=\left[\begin{array}{ccc}10,000& 20,000& 30,000\\ 50,000& 30,000& 10,000\end{array}\right]\begin{array}{c}\text{Ramkishan}\\ \text{Gurcharan Singh}\end{array}$
October Sales (in Rupees)
$\begin{array}{ccccc}& & \text{Basmati}& \text{Permal}& \text{Naura}\end{array}$
$B=\left[\begin{array}{ccc}5,000& 10,000& 6,000\\ 20,000& 10,000& 10,000\end{array}\right]\begin{array}{c}\text{Ramkishan}\\ \text{Gurucharan}\end{array}$
$\left(i\right)$ Find the combined sales in September and Octomber for each farmer in each variety.
$\left(ii\right)$ Find the decrease in sales from September to October.
$\left(iii\right)$ If both farmers receive $2$ profit on gross sales, compute the profit for each farmer and for each variety sold in October.

Q. Total number of matrices that can be formed using all 5 different letters such that no letter is repeated in any matrix is 2.5! THE above statement is true or false?

Q. Let $A=\left[\begin{array}{cc}2& -1\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}5& 2\\ 7& 4\end{array}\right],C=\left[\begin{array}{cc}2& 5\\ 3& 8\end{array}\right]$, find a matrix $D$ such that $CD-AB=0$

Q.

If (1, 2, 3) B=(3, 4), then the order of B is

Q. Let $A$ and $B$ are two square matrices such that the trace of $AB=30,$ then the trace of $BA$ is equal to

1. $\frac{1}{30}$
2. $30$
3. $-30$
4. Cannot be determined

Q. Let A be a square matrix of order two such that $A-A{A}^T=I_2$. If trace $\left(A\right)\ne R$ and entries are complex numbers, number of such matrices possible are

Q. If $A^2=3A-2I$ and $A^8=aA+bI$, then $a+b$ is

1. 7
2. -1
3. 1
4. \N

Q. If $x\left[\begin{array}{c}2\\ 3\end{array}\right]+y\left[\begin{array}{c}-1\\ 1\end{array}\right]=\left[\begin{array}{c}10\\ 5\end{array}\right]$, find the value of x.

Q. If $A^2=3A-2I$ and $A^8=aA+bI$, then $a+b$ is

1. 7
2. -1
3. 0
4. 1

Q. The number of all possible matrices of order $3\ast 3$ with each entry $0$ or $1$ is

1. $9$
2. $18$
3. $27$
4. $512$

Q. If $A(\ne O)$ is a skew-symmetric matrix of order $2\times 2$ which is formed with $0$ and $i,$ then the determinant of matrix is
(where $i=\sqrt{-1}$)

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

Q. Solve the equation for x , y , z and t if

Q.

Assume X, Y, Z, W and P are matrices of order, and respectively. If n = p, then the order of the matrix is

A p × 2 B 2 × n C n × 3 D p × n

Q. Ho w to solve t+3/2=1/5

Q. The number of permutations that can be made out of the letter of the word “ENTRANCE” so that the two ‘N’ s are always together is

1. 
2. 
3. 
4. 

Q. 34. Find the determinant of. b+c. a. c b. c+a. b c. c. a+b

Q. If $a>0$ and the discriminant of $a{x}^2+2bx+c$ is negative, then

$\left|\begin{array}{ccc}a& b& ax+b\\ b& c& bx+c\\ ax+b& bx+c& 0\end{array}\right|$ is

1. positive
2. equal to $(ac-b^2)(a{x}^2+2bx+c)$
3. negative
4. equal to $0$

Q.

Find the value of x, y and z from the following equations:
$\left(i\right)\left[\begin{array}{cc}4& 3\\ x& 6\end{array}\right]$

$\left(ii\right)\left[\begin{array}{cc}x+2& 2\\ 5+z& xy\end{array}\right]=\left[\begin{array}{cc}6& 2\\ 5& 8\end{array}\right]$

$\left(iii\right)\left[\begin{array}{c}x+y+z\\ x+z\\ y+z\end{array}\right]=\left[\begin{array}{c}9\\ 5\\ 7\end{array}\right]$

Q. If Tr(A)=[2+i] then Tr[(2-i)A]=

(This lesson is from matrix not linear programming)

Q.

Solve the following to find the value of the variables.

$\text{2y -3 = 3}$.

Q.

The real part of $\cos h\left(\alpha +i\beta \right)$is

1. $\cos h\left(\alpha \right)\cos \left(\beta \right)$

2. $\cos \left(\alpha \right)\cos \left(\beta \right)$

3. $\cos \left(\alpha \right)\cos h\left(\beta \right)$

4. $\sin \left(\alpha \right)\sin h\left(\beta \right)$

Q. If P is a 2 × 2 matrix such that PT = 3P + I, then tr (P) is equal to

Q. Let $A+2B=\left[\begin{array}{ccc}1& 2& 0\\ 6& -3& 3\\ -5& 3& 1\end{array}\right]$ and $2A-B=\left[\begin{array}{ccc}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right].$ Then $\text{tr}\left(A\right)-\text{tr}\left(B\right)$ has the value equal to

1. $0$
2. $1$
3. $2$
4. none

Q.

If $A=\left[\begin{array}{cc}0& -tan\frac{\alpha }{2}\\ tan\frac{\alpha }{2}& 0\end{array}\right]$ and I is the identity matrix of order 2, show that I + A $=(I-A)\left[\begin{array}{cc}cos\alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{array}\right]$

Q. Given $3\left[\begin{array}{cc}x& y\\ z& w\end{array}\right]=\left[\begin{array}{cc}x& 6\\ -1& 2w\end{array}\right]+\left[\begin{array}{cc}4& x+y\\ z+w& 3\end{array}\right]$, find the values of $x,y,z$ and $w$.

Q. Explain combinations.

Q. Prove the following identity:
ii.$\frac{1}{(\sin A+\cos A)}+\frac{1}{(\sin A–cosA)}=\frac{2\sin A}{(1–2{\cos }^{2}A)}$

Q. If $A=\left[\begin{array}{ccc}1& 2& -1\\ -1& 1& 2\\ 2& -1& 1\end{array}\right]$ then det (adj(adjA)) is

1. $(14{)}^1$
2. $(14{)}^2$
3. $(14{)}^3$
4. $(14{)}^4$