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

1. $27$
2. $18$
3. $81$
4. $512$

Q.

Find the value of $x,y$ and $z$ from the following equation:
(i) $\left[\begin{array}{cc}4& 3\\ x& 5\end{array}\right]=\left[\begin{array}{cc}y& z\\ 1& 5\end{array}\right]$

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

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

Q. Construct a $3\times 4$ matrix, whose elements are given by
(i) $a_{ij}=\frac{1}{2}|-3i+j|$
(ii) $a_{ij}=2i-j$

Q. Find the value of $a,b,c$ and $d$ from the equation:
$\left[\begin{array}{cc}a-b& 2a+c\\ 2a-b& 3c+d\end{array}\right]=\left[\begin{array}{cc}-1& 5\\ 0& 13\end{array}\right]$

Q. If a matrix has $18$ elements , what are the possible orders it can have? what, if it has $5$ elements?

Q. If a matrix has $24$ elements, what are the possible order it can have? What, if it has $13$ elements?

Q.

In the matrix, write :
$A=\left[\begin{array}{c}2\\ 35\\ \sqrt{3}\end{array}\begin{array}{c}5\\ -2\\ 1\end{array}\begin{array}{c}19\\ \frac{5}{2}\\ -5\end{array}\begin{array}{c}-7\\ 12\\ 17\end{array}\right]$

(i) The order of the matrix
(ii) The number of elements
(iii) Write the elements $a_{13},a_{21},a_{33},a_{24},a_{23}$

Q. Which of the given values of $x$ and $y$ make the following pair of matrices equal.
$\left[\begin{array}{cc}3x+7& 5\\ y+1& 2-3x\end{array}\right]=\left[\begin{array}{cc}0& y-2\\ 8& 4\end{array}\right]$

1. Not possible to find
2. $x=\frac{-1}{3},y=\frac{-2}{3}$
3. $y=7,x=\frac{-2}{3}$
4. $x=\frac{-1}{3},y=7$

Q. Construct a $2\times 2$ matrix, $A=\left[a_{ij}\right]$, whose elements are given by:
(i) $a_{ij}=\frac{(i+j{)}^2}{2}$
(ii) $a_{ij}=\frac{i}{j}$
(iii) $a_{ij}=\frac{(i+2j{)}^2}{2}$

Q. Find $X$ and $Y$ , if $2\left[\begin{array}{cc}1& 3\\ 0& x\end{array}\right]+\left[\begin{array}{cc}y& 0\\ 1& 2\end{array}\right]=\left[\begin{array}{cc}5& 6\\ 1& 8\end{array}\right]$

Q. If $A=\left[\begin{array}{c}1\\ 5\\ 1\end{array}\begin{array}{c}2\\ 0\\ -1\end{array}\begin{array}{c}-3\\ 2\\ 1\end{array}\right],B=\left[\begin{array}{c}3\\ 4\\ 2\end{array}\begin{array}{c}-1\\ 2\\ 0\end{array}\begin{array}{c}2\\ 5\\ 3\end{array}\right]$ and $C=\left[\begin{array}{c}4\\ 0\\ 1\end{array}\begin{array}{c}1\\ 3\\ -2\end{array}\begin{array}{c}2\\ 2\\ 3\end{array}\right]$, then compute $(A+B)$ and $(B-C)$.

Q. Simplify: $\cos \theta \left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]+\sin \theta \left[\begin{array}{cc}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{array}\right]$

Q. Find $X$, if $Y=\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right]$ and $2X+Y=\left[\begin{array}{cc}1& 0\\ -3& 2\end{array}\right]$

Q. If $A=\left[\begin{array}{c}\frac{2}{3}\\ \frac{1}{3}\\ \frac{7}{3}\end{array}\begin{array}{c}1\\ \frac{2}{3}\\ 2\end{array}\begin{array}{c}\frac{5}{3}\\ \frac{4}{3}\\ \frac{2}{3}\end{array}\right]$ and $B=\left[\begin{array}{c}\frac{2}{5}\\ \frac{1}{5}\\ \frac{7}{5}\end{array}\begin{array}{c}\frac{3}{5}\\ \frac{2}{5}\\ \frac{6}{5}\end{array}\begin{array}{c}1\\ \frac{4}{5}\\ \frac{2}{5}\end{array}\right]$, then compute $3A-5B$.

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 values of $x$ and $y$.

Q. Compute the indicated products
(i) $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$

Q. Find $X$ and $Y$, if
(i) $X+Y=\left[\begin{array}{cc}7& 0\\ 2& 5\end{array}\right]$ and $X-Y=\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$
(ii) $2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]$ and $3X+2Y=\left[\begin{array}{cc}2& -2\\ -1& 5\end{array}\right]$

Q. Let $A=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right],B=\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right],C=\left[\begin{array}{cc}-2& 5\\ 3& 4\end{array}\right]$
Find each of the following
(i) $A+B$
(ii) $A-B$
(iii) $3A-C$
(iv) $AB$
(v) $BA$

Q. Compute the following:
(i) $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]+\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$
(ii) $\left[\begin{array}{cc}{a}^2+b^2& b^2+c^2\\ a^2+c^2& a^2+b^2\end{array}\right]+\left[\begin{array}{cc}2ab& 2bc\\ -2ac& -2ab\end{array}\right]$
(iii) $\left[\begin{array}{c}-1\\ 8\\ 2\end{array}\begin{array}{c}4\\ 5\\ 8\end{array}\begin{array}{c}-6\\ 16\\ 5\end{array}\right]+\left[\begin{array}{c}12\\ 8\\ 3\end{array}\begin{array}{c}7\\ 0\\ 2\end{array}\begin{array}{c}6\\ 5\\ 4\end{array}\right]$
(iv) $\left[\begin{array}{cc}{\cos }^{2}x& {\sin }^{2}x\\ {\sin }^{2}x& {\cos }^{2}x\end{array}\right]+\left[\begin{array}{cc}{\sin }^{2}x& {\cos }^{2}x\\ {\cos }^{2}x& {\sin }^{2}x\end{array}\right]$

Q. Solve the equation for $x,y,z$ and $t$ if $2\left[\begin{array}{cc}x& z\\ y& t\end{array}\right]+3\left[\begin{array}{cc}1& -1\\ 0& 2\end{array}\right]=3\left[\begin{array}{cc}3& 5\\ 4& 6\end{array}\right]$