Matrix (Myanmar Exam Board)

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Group (2015-2019)

1. (2015/Myanmar /q10b )
Given that the matrix $A=\left(\begin{array}{rr}2 & -1 \\ -3 & 4\end{array}\right)$ and that $A^{2}-k A+5 I=O$, find the $\begin{array}{lll}\text { value of } k . & \cdot(5 \text { marks })\end{array}$

2. (2015/Myanmar /q11a )
Find the solution set of the systems of equations $\left.\begin{array}{l}5 x+6 y=25 \\ 3 x+4 y=17\end{array}\right\}$ by matrix method; the variables are on the set of real numbers.

3. (2015/FC /q10b )
Given that $A=\left(\begin{array}{cc}4 & 3 \\ 1 & 1\end{array}\right)$ and $B=\left(\begin{array}{rr}4 & 2 \\ -5 & -3\end{array}\right)$, write down the inverse matrix $A^{-1}$ and find the matrices $P$ and $Q$ such that $P A=2 I$ and $A Q=2 B$. (5 marks)

4. (2015/FC /q11a )
Find the solution set of the system of equations $2 x-5 y=1$ $3 x-7 y=2$ by matrix method. $\quad(5$ marks $)$

5. (2016/Myanmar /q10b )
The matrices $A$ and $B$ are such that $A=\left(B^{-1}\right)^{2}$. Given that $B=\left(\begin{array}{cc}2 & -1 \\ 2 & 1\end{array}\right)$ find the value of the constant $k$ for which $k B^{-1}=4 A+I$, where $I$ is the identity matrix of order 2 .

6. (2016/Myanmar /q11a )
Given that $A=\left(\begin{array}{cc}4 & -1 \\ -3 & 2\end{array}\right)$, use the inverse matrix of $A$ to solve the simultaneous equations $y-4 x+8=0,2 y-3 x+1=0$.

7. (2016/FC /q10b )
Find the inverse of the matrix $M=\left(\begin{array}{ll}3 & 5 \\ 1 & 2\end{array}\right)$ and investigate whether or not the squares of $M$ and $M^{1}$ are inverses of each other.

8. (2016/FC /q11a )
Find the inverse of the matrix $\left(\begin{array}{ll}4 & 3 \\ 7 & 6\end{array}\right)$, and use it to solve the system of equations $3 y+4 x+7=0$ and $14 x+12 y+32=0$

9. (2017/Myanmar /q10b )
Given that $A=\left(\begin{array}{lr}4 & 1 \\ -9 & -2\end{array}\right)$ and $B=\left(\begin{array}{cc}-3 & 1 \\ -1 & 2\end{array}\right)$. Solve the equation $A X=2 B-A^{2}$
Q10 (b) Solution

10. (2017/Myanmar /q11a )
Find the inverse of the matrix $\left(\begin{array}{cc}1 & 1 \\ -1 & 1\end{array}\right)$ and use it to solve the following system of equations, $y-x=1$ and $x+y=3$
Q11(a) Solution

11. (2017/FC /q10b )
If $\mathrm{ps} \neq \mathrm{qr}$, find the $2 \times 2$ matrix $\mathrm{X}$ such that $\left(\begin{array}{ll}\mathrm{p} & \mathrm{q} \\ \mathrm{r} & \mathrm{s}\end{array}\right) \mathrm{X}=\left(\begin{array}{ll}\mathrm{q} & \mathrm{p} \\ \mathrm{s} & \mathrm{r}\end{array}\right)$. Find also $\mathrm{X}^{-1}$, if it exists. $\quad(5 \mathrm{marks})$

12. (2017/FC /q11a )
Find the inverse of the matrix $\left(\begin{array}{ll}7 & 4 \\ 3 & 2\end{array}\right)$, and use it to find the solution set of the system of equations $\quad 7 x+4 y=16$ $2 y+3 x=6 . \quad(5$ marks $)$

13. (2018/Myanmar /q10b )
Given that $A=\left(\begin{array}{cc}5 & 1 \\ a+1 & a\end{array}\right)$ and $\operatorname{det} A=7$, find the value of $a$ and then calculate the values of $x$ and $y$ such that $A^{2}-x A^{-1}-y I=O$, where $I$ is the unit matrix order 2 .
Click for Solution

14. (2018/Myanmar /q11a )
Find the inverse of the matrix $\left(\begin{array}{rr}7 & -4 \\ -3 & 2\end{array}\right)$ and use it to solve the following systems: $7 x-4 y=13,2 y-3 x=-5$
Click for Solution

15. (2018/FC /q10b )
Solve the matrix equation $\left(\begin{array}{lr}-2 & 3 \\ 1 & -4\end{array}\right) X=\left(\begin{array}{cc}-3 & -5 \\ -16 & -20\end{array}\right)$, Hence find $x$ and $y$, if $X=\left(\begin{array}{lr}x+2 y & 16 \\ 7 & 2 x-y\end{array}\right)$

16. (2018/FC /q11a )
Find the inverse of the matrix $\left(\begin{array}{ll}4 & 3 \\ 7 & 6\end{array}\right)$, and use it to solve the system of equations $$\begin{aligned}&7 x+6 y+16=0 \\&3 y+4 x+7=0\end{aligned}$$

17. (2019/Myanmar /q3a )
The matrices $\mathrm{A}=\left(\begin{array}{ll}2 & 0 \\ 0 & 5\end{array}\right)$ and $\left(\begin{array}{ll}\mathrm{x} & \mathrm{y} \\ 0 & \mathrm{z}\end{array}\right)$ are such that $\mathrm{AB}=\mathrm{A}+\mathrm{B}$. Find the values of $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ Click for Solution

18. (2019/Myanmar /q9b )
Using the definition of inverse matrix, find the inverse of the matrix $\left(\begin{array}{ll}3 & 1 \\ 2 & 1\end{array}\right) .(5$ marks $)$ Click for Solution

19. (2019/Myanmar /q10a )
Find the inverse of matrix $\left(\begin{array}{ll}5 & 6 \\ 7 & 8\end{array}\right)$. Use it to determine the coordinates of the point of intersection of the lines $5 x+6 y=7$ and $8 y+7 x=10 . \quad$ (5 marks) Click for Solution

20. (2019/FC /q3a )
Given that $2\left(\begin{array}{rr}1 & 4 \\ -6 & 3\end{array}\right)+\left(\begin{array}{rr}1 & 2 \\ -1 & 3\end{array}\right)\left(\begin{array}{rr}3 & 4 \\ 5 & -4\end{array}\right)=\left(\begin{array}{cc}a & b \\ 0 & c\end{array}\right)$, find the values of $a, b$ and $c$.(3 marks) Click for Solution 3(a)


21. (2019/FC /q9b )
Using the definition of inverse matrix. find the inverse of the matrix $\left(\begin{array}{ll}3 & 5 \\ 2 & 2\end{array}\right)$. (5 marks ) Click for Solution 9(b)


22. (2019/FC /q10a )
Find the inverse of the matrix $\left(\begin{array}{ll}3 & 4 \\ 2 & 6\end{array}\right)$. Use it to determine the coordinates of intersection of the lines $3 x+4 y=18$ and $6 y+2 x=22$. (5 marks) Click for Solution 10(a)

Answer (2015-2019)

1.  $k=6$
2.  $\{(-1,5)\}$  
3.  $A^{-1}=\left(\begin{array}{cc}1 & -3 \\ -1 & 4\end{array}\right)$$P=\left(\begin{array}{cc}2 & -6 \\ -2 & 8\end{array}\right)$$\left(\begin{array}{cc}38 & 22 \\ -48 & -28\end{array}\right)$
4.  $\{(3,1)\}$
5.  $k=3$
6.  $x=3, y=4$ 
7.  $M^{-1}=\left(\begin{array}{cc}2 & -5 \\ -1 & 3\end{array}\right)$ (Yes)
8.  $\left(\begin{array}{cc}2 & -1 \\ -\frac{7}{3} & \frac{4}{3}\end{array}\right), x=2, y=-5$
9.  $X=\left(\begin{array}{cc}10 & -9 \\ -53 & 36\end{array}\right)$
10. $x=1, y=2$
11. $X=\left(\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right)$, $X^{-1}=\left(\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right)$
12.  $\left(\begin{array}{cc}1 & -2 \\ -\dfrac 32 & \dfrac 72\end{array}\right)$ $(4,-3)$ 
13.  $a=2, x=-49, y=42$
14.  $\left(\begin{array}{ll}1 & 2 \\ \frac{3}{2} & \frac{7}{2}\end{array}\right), x=3, y=2$
15.  $x=\left(\begin{array}{cc}12 & 16 \\ 7 & 9\end{array}\right), x=6, y=3$
16.  $\left(\begin{array}{cc}2 & -1 \\ -\frac{7}{3} & \frac{4}{3}\end{array}\right), x=2, y=-5$
17.  $x=2,y=0,z=\frac 54$
18.  $A^{-1}=\left(\begin{array}{cc}1&-1\\-2&3\end{array}\right)$
19.   $A^{-1}=\left(\begin{array}{cc}-4&3\\3.5&-2.5\end{array}\right), (2,-0.5)$
20.  $a=15,b=4,c=-10$ 
21.  $A^{-1}=\left(\begin{array}{ll}-\dfrac 12 & \dfrac 54\\ \dfrac 12& -\dfrac 34\end{array}\right)$
22.  $A^{-1}=\left(\begin{array}{ll}\dfrac 35 & -\dfrac 25\\ -\dfrac 15& \dfrac{3}{10}\end{array}\right)$ , (2,3)


Group (2014)

1. Given that $P=\left(\begin{array}{rr}2 & -3 \\ -2 & 1\end{array}\right)$ and that $P^{2}-12 P^{-1}+k I=0$, where $I$ is the unit matrix of order $2, \mathrm{fi}$ nd the value of $k$. (5 marks)

2. If $A=\left(\begin{array}{rr}1 & -1 \\ 2 & 1\end{array}\right), B=\left(\begin{array}{rr}a & 1 \\ b & -1\end{array}\right)$ and $(A+B)^{2}=A^{2}+B^{2}+2 A B$, then find the values of $a$ and $b$. (5 marks)

3. If $A=\left(\begin{array}{rr}5 & -4 \\ -1 & 5\end{array}\right), \mathrm{I} \mathrm{i}^{3}$ the unit matrix of order 2 and $A^{2}-10 A+k I=0$, find the value of $k$. Show alse that $(A-7 I)(A-3 I)=O$. (5 marks)

4. Given that $A=\left(\begin{array}{ll}2 & 1 \\ 3 & 1\end{array}\right)$ and $\mathrm{B}=\left(\begin{array}{ll}2 & 3 \\ 3 & 5\end{array}\right)$, find $\left(A^{\prime}+B^{-1}\right)(B-2 A)$. (5 marks)

5. Solve the matrix equation $\left(\begin{array}{ll}3 & 1 \\ 3 & 2\end{array}\right) X-2\left(\begin{array}{ll}0 & 9 \\ 2 & 5\end{array}\right)=\left(\begin{array}{cc}7 & -8 \\ 2 & 16\end{array}\right)$. (5 marks)

6. Solve the matrix equation $P\left(\begin{array}{lr}1 & -1 \\ 0 & 1\end{array}\right)+2\left(\begin{array}{ll}2 & 3 \\ 4 & 1\end{array}\right)=\left(\begin{array}{ll}5 & 7 \\ 11 & 2\end{array}\right)$ for $2 \times 2$ matrix $P$. (5 marks)

7. Given that $A=\left(\begin{array}{ll}2 & 1 \\ 5 & 3\end{array}\right)$ and $B=\left(\begin{array}{ll}3 & 2 \\ 2 & 1\end{array}\right)$, write down the inverse matrix of $A$ and use it to solve the equation $X A=3 B+2 A$. (5 marks)

8. The matrices $A$ and $B$ are given by $A=\left(\begin{array}{cc}-2 & 3 \\ 1 & 0\end{array}\right), B=\left(\begin{array}{rr}5 & 1 \\ -1 & 2\end{array}\right)$. Find matrices $P$ and $Q$ such that $P=2 A+B^{2}$ and $A Q+B Q=I$. (5 marks)

9. Given that $A=\left(\begin{array}{rr}3 \frac{1}{2} & -1 \frac{1}{2} \\ -2 & 1\end{array}\right), B=\left(\begin{array}{ll}7 & 6 \\ 6 & 5\end{array}\right), C=\left(\begin{array}{ll}3 & 2 \\ 1 & 4\end{array}\right)$, find $A^{-1}$ and $B^{-1}$ and use the result to find the matrix $X$ such that $B X A=C$.  (5 marks)

10. If $p s \neq q r$, find $2 \times 2$ matrix $X$ such that $\left(\begin{array}{cc}p & q \\ r & s\end{array}\right) X=\left(\begin{array}{cc}q & p \\ s & r\end{array}\right)$. Find also $X^{-1}$, if it exists. (5 marks)

11. Use the definition of inverse of matrix, to find the inverse of $\left(\begin{array}{rr}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right)$. (5 marks)

12. Given that $A=\left(\begin{array}{rr}2 & 1 \\ -5 & -4\end{array}\right)$ and $B=\left(\begin{array}{rr}3 & -1 \\ -1 & 1\end{array}\right)$. Verify that $(A B)^{-1}=B^{-1} A^{-1}$. (5 marks)

13. By using a matrix method, find the solution set of the system of equations$$\begin{aligned}&6 x+3 y=15 \\&2 y-3 x=-18\end{aligned}$$ (5 marks)

14. Solve the following system of equations by matrix method.$$\begin{aligned}&4 x+2 y=3 \\&3 x-4 y=5\end{aligned}$$ (5 marks)

15. Solve the following system of equations by matrix method:$$\begin{aligned}&3 x+7 y=23 \\&y+x=5\end{aligned}$$ (5 marks)

16. Find the solution set of the system of equations $3 x-7 y=35, x+y=5$, by matrix method; the variables are on the set of real numbers. (5 marks)

17. Use the matrix method to find the solution set of the system of linear equations$$\begin{aligned}&3 y-2 x=1 \\&x+2 y=10\end{aligned}$$ (5 marks)

18. Find the inverse of the matrix $\left(\begin{array}{rr}9 & -2 \\ 2 & 3\end{array}\right)$. Hence determine the coordinates of the point of intersection of the lines $9 x-2 y-13=0$ and $2 x+3 y+4=0$. (5 marks)

19. Find the inverse of the matrix $\left(\begin{array}{ll}a & b \\ 1 & 2\end{array}\right)$ where $2 a \neq b$ and use it to solve the simultaneous equation $a x+b y=2 a^{2}$ and $x+2 y=b$ in terms of $a$ and $b$. (5 marks)

20. Try to solve $2 x+y=2$ and $6 x+3 y=-2$ by matrices. Explain with the aid of Cartesian diagram, why you failed. (5 marks)

21. Try to solve the system of equations $2 x+y=2,6 x+3 y=-2$. Explain with the aid of cartesians diagram, why you failed. (5 marks)

Answer (2014)

1. $ k=-13$
2. $a=-1$ and $b=-2$
3. $k=21$
4. $\left(\begin{array}{ll}-14 & 7 \\ -5 & 7\end{array}\right) \quad $
5. $\left(\begin{array}{ll}\frac{8}{3}&0\\\frac{16}{3} & 10\end{array}\right)$
6. $\left(\begin{array}{ll}1 & 2 \\ 3 & 3\end{array}\right)$
7. $\left(\begin{array}{rr}3 & -1 \\ -5 & 2\end{array}\right),\left(\begin{array}{rr}-1 & 3 \\ 3 & 2\end{array}\right)$
8. $p=\left(\begin{array}{cc}20 & 13 \\ -5 & 3\end{array}\right), Q=\left(\begin{array}{cc}\frac{1}{3}&-\frac{2}{3} \\ 0 & \frac{1}{2}\end{array}\right)$
9. $A^{-1}=\left(\begin{array}{ll}2 & 3 \\ 4 & 7\end{array}\right), B^{-1}=\left(\begin{array}{rr}-5 & 6 \\ 6 & -7\end{array}\right), X=\left(\begin{array}{rr}38 & 71 \\ -42 & -79\end{array}\right) \quad$ 
10. $X^{-1}=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$
11. $\left(\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right) $
12. Verify
13. $\{(4,-3)\}$
14. $x=1, y=-\frac{1}{2}$
15. $x=3, y=2$
16. $\{(7,-2)\}$
17. $\{(4,3)\} \quad$ 
18. $\left(\begin{array}{rr}\frac{3}{31} & \frac{2}{31} \\ -\frac{2}{31} & \frac{9}{31}\end{array}\right),(1,-2)$
19. $\frac{1}{2 a-b}\left(\begin{array}{rr}2 & -b \\ -1 & a\end{array}\right), x=2 a+b, y=-a$
20. There is no point of intersection of two straight lines. $\therefore$ The system has no solution.
21. There is no point of intersection of two straight lines. $\therefore$ The system has no solution.


Group (2013)

1. Given that $A=\left(\begin{array}{cc}a+3 & -4 \\ -1 & 5\end{array}\right), I$ is the unit matrix of order 2 and $A^{2}-9 A+16 I=0$, find the value of $a$. Find also the transpose of $A$. (5 marks)

2. Given that $\mathrm{A}=\left(\begin{array}{ll}2 & 3 \\ c & 2\end{array}\right), \operatorname{det} A=-5$, find $c .$ Hence verify that $A^{2}-4 A-5 I=0$ when $I$ is unit matrix of order 2 . (5 marks)

3. Given that $A=\left(\begin{array}{cc}5 & 1 \\ a+1 & a\end{array}\right)$ and $\operatorname{det} A=7$, find the value of $a$. If $I$ is the unit matrix of order 2 , verify that $A^{2}-7 A+7 I=0$. (5 marks)

4. If $A=\left(\begin{array}{ll}2 & 1 \\ 5 & 3\end{array}\right)$ and $B=\left(\begin{array}{rr}3 & 4 \\ -2 & 1\end{array}\right)$, find $(3 A-2 I)\left(2 A^{-1}+B^{\prime}\right)$ where $I$ is the unit matrix of order 2 . (5 marks)

5. Given that $X=\left(\begin{array}{ll}0 & 1 \\ 2 & 3\end{array}\right)$ and $Y=\left(\begin{array}{cc}3 & 2 \\ 1 & 0\end{array}\right)$, find out whether or not $(X+Y)(X-Y)=X^{2}-Y^{2}$. (5 marks)

6. Solve the equation for $2 \times 2$ matrix $X, 2\left(\begin{array}{cr}1 & 3 \\ 2 & -1\end{array}\right)+\left(\begin{array}{rc}1 & -1 \\ -2 & 3\end{array}\right) X=\left(\begin{array}{cc}2 & 5 \\ 3 & -1\end{array}\right)$. (5 marks)

7. Let $A=\left(\begin{array}{rr}3 & 4 \\ -2 & 1\end{array}\right)$ and $B=\left(\begin{array}{rr}2 & 5 \\ -1 & 3\end{array}\right)$ be given. Find $A^{-1}$ and use it to solve the equation $X A=3 B^{\prime}-2 A$. (5 marks) (5 marks)

8. Given $P=\left(\begin{array}{rr}2 & -1 \\ 7 & 4\end{array}\right), Q=\left(\begin{array}{r}5 \\ -1\end{array}\right)$ and $R=\left(\begin{array}{c}10 \\ 9\end{array}\right)$, write down the inverse of the matrix $P$ and use it to find the matrix $X$ in the matrix equation $P X+Q=R$. (5 marks)

9. By using a matrix method, find the solution set of the system of equations$$\begin{aligned}&5 x+2 y=11 \\&4 x-3 y=18\end{aligned}$$. (5 marks)

10. Use the matrix method to find the solution set of the system of equations: $3 x-7 y=44$ and $8 y+2 x+34=0$. (5 marks)

11. Find the solution set of the equations:$2 x-5 y=-1$ $y-3 x=-5$, by the matrix method. (5 marks)

12. Find the solution set of the system of equations by matrix method:$$\begin{aligned}&3 x+y=10 \\&y-x=-2\end{aligned}$$. (5 marks)

13. Given that $A=\left(\begin{array}{ll}4 & 7 \\ 3 & 4\end{array}\right)$, find $A^{-1}$ and use it to solve the equations $4 x+7 y=-2,3 x+4 y=1$. (5 marks)

14. Find the inverse of the matrix $A=\left(\begin{array}{ll}7 & 8 \\ 5 & 6\end{array}\right)$ and use it to solve the systems $7 x+8 y=10,5 x+6 y=7$. (5 marks)

15. Write down the inverse matrix of the matrix $\left(\begin{array}{rr}9 & -2 \\ 2 & 3\end{array}\right)$ and use it to solve the following simultaneous equations $9 x-2 y-13=0$ and $2 x+3 y+4=0$. (5 marks)

16. Find the inverse of the matrix $\left(\begin{array}{rr}1 & 2 \\ -3 & 1\end{array}\right)$ and use it to find the solution set of the system of equations$$\begin{aligned}&x+2 y=8 \\&y-3 x=-3\end{aligned}$$. (5 marks)

17. Find the inverse of the matrix $A=\left(\begin{array}{rr}1 & 3 \\ -2 & 5\end{array}\right)$ and use it to find the solution set of the system of equations $x+3 y=7$ and $5 y-2 x=-3$. (5 marks)

18. Find the inverse of the matrix $\left(\begin{array}{rr}2 & -3 \\ -1 & 5\end{array}\right)$ and use it to find the solution set of the system of equations $5 y-x=3$,$$2 x-3 y=1$$ (5 marks)

19. Find the inverse of the matrix $\left(\begin{array}{ll}3 & 4 \\ 2 & 6\end{array}\right)$. Hence, determine the coordinates of the point of intersection of the lines $3 x+4 y=18$ and $2 x+6 y=22$. (5 marks)

20. Show that for $(A B)^{-1}=B^{-1} A^{-1}$ for $A=\left(\begin{array}{ll}3 & 4 \\ 2 & 3\end{array}\right)$ and $B=\left(\begin{array}{ll}3 & 7 \\ 2 & 5\end{array}\right)$. (5 marks)

21. If $A=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$, show that $A^{2}-(2 \cos \theta) A+I=O$ where $I$ is the unit matrix of order 2 . (5 marks)

22. Show that $A=\left(\begin{array}{rc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right)$ satisfies $A^{2}+I=2 A \cos \alpha$ where $I$ is the unit matrix of order 2 . (5 marks)

Answer (2013)
1. $a=1, A^{\prime}=\left(\begin{array}{rr}4 & -1 \\ -4 & 5\end{array}\right) \quad$ 
2. $c=3 \quad$ 
3. $a=2 $
4. $\left(\begin{array}{cc}18 & -1 \\ 93 & -25\end{array}\right) \quad $
5. $(X+Y)(X-Y) \neq X^{2}-Y^{2} \quad$ 
6. $\left(\begin{array}{ll}-1 & -2 \\ -1 & -1\end{array}\right) \quad $
7. $\left(\begin{array}{rr}-2 & -3 \\ 3 & -5\end{array}\right)$ 
8. $\left(\begin{array}{c}2 \\ -1\end{array}\right) \quad$ 
9. $\{(3,-2)\}$
10. $\{(3,-5)\} \quad$ 
11. $\{(2,1)\} \quad$ 
12. $\{(3,1)\}$
13. $A^{-1}=\left(\begin{array}{cc}-4/5 & 7/5 \\ 3/5 & -4/5\end{array}\right)$ 
14. $A^{-1}=\left(\begin{array}{cc}3 & -4 \\ -5 / 2 & 72\end{array}\right), x=2, y=-\frac{1}{2}$
15. $\left(\begin{array}{rr}3 / 31 & 2 / 31 \\ -2 / 31 & 9 / 31\end{array}\right), x=1, y=-2$
16. $\left(\begin{array}{cc}1 / 7 & -2 / 7 \\ 3 / 7 & 1 / 7\end{array}\right),\{(2,3)\} \quad$ 
17. $A^{-1}=\left(\begin{array}{ll}5 / 11 & -3 / 11 \\ 2 / 11 & 1 / 11\end{array}\right),\{(4,1)\}$
18. $\left(\begin{array}{ll}5 / 7 & 3 / 7 \\ 1 / 7 & 2 / 7\end{array}\right),\{(2,1)\} \quad$ 
19. $\left(\begin{array}{ll}3 / 5 & -2 / 5 \\ -1 / 5 & 3 / 10\end{array}\right),(2,3)$
20. Show
21. Show
22. Show


Group (2012)



$\quad\;\,$ $\,$
1.If $A=\left(\begin{array}{cc}k & 4 \\ 1 & k\end{array}\right)$ satisfies $A^{2}-2 A-3 I=0$, find the value of $k$. (5 marks)
2.Given that $A=\left(\begin{array}{rr}2 & -3 \\ 4 & 1\end{array}\right), B=\left(\begin{array}{lc}2 & 18 \\ -24 & 8\end{array}\right)$ and $A^{2}+k A+B=0$, then find the value of $k$. (5 marks)
3.Let $A=\left(\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right)$, find $p, q$ such that $A^{2}=p A+q I$, where $I$ is the unit matrix of order 2. (5 marks)
4.Given that $M=\left(\begin{array}{cc}k & 4 \\ 1 & 6\end{array}\right) . I$ is the unit matrix of order 2 and $M^{2}-9 M+(4 k+2) I=0$, find the value of the number $k$. Find also the inverse of $M$. (5 marks)
5.If $A=\left(\begin{array}{ll}3 & -4 \\ 1 & -2\end{array}\right)$, find $A^{2}+2 A^{-1}+3 I$ where $I$ is the unit matrix of order 2 . (5 marks)
6.If $A=\left(\begin{array}{ll}-2 & 3 \\ -3 & 4\end{array}\right)$, prove that $A^{2}+2 A^{-1}=3 I$ where $I$ is the unit matrix of order 2. (5 marks)
7.Given that $A=\left(\begin{array}{ll}3 & 4 \\ 2 & 3\end{array}\right)$. Verify that $\left(A^{2}\right)^{-1}=\left(A^{-1}\right)^{2}$. (5 marks)
8.If $X$ is a $2 \times 2$ matrix such that $\left(\begin{array}{ll}3 & 1 \\ 3 & 2\end{array}\right) X=\left(\begin{array}{ll}0 & 7 \\ 9 & 2\end{array}\right)$, find $X$ and its transpose. (5 marks)
9.Given that $A=\left(\begin{array}{ll}2 & 1 \\ 5 & 3\end{array}\right)$ and $B=\left(\begin{array}{ll}3 & 4 \\ -2 & 1\end{array}\right)$, write down the matrix $A^{-1}$ and use it to solve the equation $A X=B-A$. (5 marks)
10.Let $A=\left(\begin{array}{ll}3 & 4 \\ -2 & 1\end{array}\right)$ and $B=\left(\begin{array}{rr}2 & -1 \\ 5 & 3\end{array}\right)$. Write down the inverse matrix of $A$ and use it to solve the equation $X A=3 B-2 A$. (5 marks)
11.Given that $A=\left(\begin{array}{ll}2 & 1 \\ 5 & 3\end{array}\right)$ and $B=\left(\begin{array}{ll}3 & 4 \\ -2 & 1\end{array}\right)$, write down the inverse matrix of $A$ and use it to solve the equation $X A=3 B+2 A$. (5 marks)
12.Given that $A=\left(\begin{array}{ll}7 & 5 \\ 8 & 9\end{array}\right)$ and $B=\left(\begin{array}{ll}3 & 5 \\ 1 & 2\end{array}\right)$, write down the inverse matrix of $B$ and use it to find the matrix $P$ such that $P B=A$. (5 marks)
13.Find the $2 \times 2$ matrix $X$ in the equation $X\left(\begin{array}{cc}2 & -2 \\ 3 & -5\end{array}\right)=3\left(\begin{array}{cc}2 & -4 \\ 1 & 3\end{array}\right)-2\left(\begin{array}{cc}5 & -4 \\ 4 & 7\end{array}\right)$. (5 marks)
14.Let $A=\left(\begin{array}{cc}2 & 3 \\ 1 & 1\end{array}\right)$ and $B=\left(\begin{array}{lr}2 & 1 \\ -5 & -3\end{array}\right) .$ Solve for $2 \times 2$ matrix $X$ such that $A X=2 A^{\prime}+5 B^{-1}$. (5 marks)
15.Find the solution set of the equations:$$\begin{aligned}&2 x-5 y=7 \\&3 x-4 y=14, \text { using by matrix method. }\end{aligned}$$ (5 marks)
16.Find the solution set of the system of equations $3 x+2 y=7$ and $5 x-y=3$ by matrix method. (5 marks)
17.Find the solution set of the system of equations by matrix method:$$3 x+5 y=21,4 x+3 y=17$$ (5 marks)
18.Find the solution set of the system of equations $$\begin{aligned}&2 x+y=55 \\&2 y-x=10 \text { by matrix method. }\end{aligned}$$ (5 marks)
19.Find the solution set of the following system of equations by matrix method: $x+3 y=7,5 y-2 x=-3$. (5 marks)
20.Find the solution set of the system of equations, $3 x-4 y=20$ and $5 y-2 x=-18$ by matrix method. (5 marks)
21.Find the solution set of the system of equations $3 x+4 y=18$ and $6 y+4 x=22$ by matrix method. (5 marks)
22.Use a matrix method to obtain, in terms of $a$ and $b$ where $2 a \neq b$, the solution set of the equations $a x+b y=2 a^{2}, x+2 y=b$. (5 marks)


Answer (2012)



$\quad\;\,$ $\,$
1.$\boldsymbol{k}=1$
2.$k=3$
3.$p=4, q=-3$
4.$k=3, M^{-1}=\left(\begin{array}{ll}\frac{3}{7} & \frac{-2}{7} \\ \frac{-1}{14} & \frac{3}{14}\end{array}\right)$
5.$\left(\begin{array}{rr}10 & -8 \\ 2 & 0\end{array}\right)$
6.Prove
7.Verify
8.$X=\left(\begin{array}{lr}-3 & 4 \\ 9 & -5\end{array}\right) ; X^{\prime}=\left(\begin{array}{lr}-3 & 9 \\ 4 & -5\end{array}\right)$
9.$A^{-1}=\left(\begin{array}{lr}3 & -1 \\ -5 & 2\end{array}\right), X=\left(\begin{array}{ll}-31 & 15 \\ -33 & 14\end{array}\right)$
10.$A^{-1}=\left(\begin{array}{lr}3 & -1 \\ -5 & 2\end{array}\right), X=\left(\begin{array}{ll}-31 & 15 \\ -33 & 14\end{array}\right)$
11.$A^{-1}=\left(\begin{array}{lr}3 & -1 \\ -5 & 2\end{array}\right), X=\left(\begin{array}{lr}10 & 11 \\ -19 & -19\end{array}\right)$
12.$B^{-1}=\left(\begin{array}{cc}2 & -5 \\ -1 & 3\end{array}\right), P=\left(\begin{array}{cc}9 & -20 \\ 7 & -13\end{array}\right) \quad$
13.$X=\left(\begin{array}{cc}-8 & 4 \\ -10 & 5\end{array}\right)$
14.$X=\left(\begin{array}{rr}-76 & -31 \\ -57 & 23\end{array}\right)$
15.$\{(6,1)\} \quad$
16.$\{(1,2)\} \quad$
17.$\{(2,3)\}$
18.$\{(20,15)\} $
19.$\{(4,1)\} $
20.$\{(4,-2)\} $
21.$\{(10,-3)\} $
22.$\{(2 a+b,-a)\}$


Group (2011)


$\quad\;\,$$\,$
1.Given that $A=\left(\begin{array}{cc}2 & -3 \\ -2, & 1\end{array}\right)$ and that $A^{2}-3 A+k I=0$, find the value of $k$. $\mbox{ (5 marks)}$

2.Given that $D=\left(\begin{array}{rr}3 & -4 \\ 1 & -1\end{array}\right)$ and that $D^{2}+2 D^{-1}-k I=0$, find the value of $k$. $\mbox{ (5 marks)}$

3.If $A=\left(\begin{array}{ll}3 & 1 \\ 5 & 2\end{array}\right), B=\left(\begin{array}{ll}4 & 1 \\ 5 & 3\end{array}\right)$ and $A^{2}+A^{-1}=m B$, where $m$ is a real number, find the value of $m$. $\mbox{ (5 marks)}$

4.Find the two matrices of the form $A=\left(\begin{array}{ll}a & 1 \\ 0 & b\end{array}\right)$ such that $A^{2}=I$, where $I$ is the unit matrix of order $2 .$ $\mbox{ (5 marks)}$

5.Given that $M=\left(\begin{array}{cc}3 & 2 a \\ -1 & 2\end{array}\right)$ and det $M=10$, find the value of $a .$ If $I$ is the unit matrix of order 2 , verify that $M^{2}-5 M+10 I=0$ $\mbox{ (5 marks)}$

6.Given that $A=\left(\begin{array}{cc}3 & 2 \\ 1 & 4\end{array}\right)$ and $B=\left(\begin{array}{rr}k & 2 k \\ 3 & 4\end{array}\right)$, find the value of $k$ for which the determinant of the matrix $A B$ is $-20 .$ Hence, find the inverse matrix of $B$. $\mbox{ (5 marks)}$

7.Find the matrix $X$ of the form $X=\left(\begin{array}{ll}x & 1 \\ 0 & y\end{array}\right)$ such that $X^{3}=\left(\begin{array}{cc}-1 & 1 \\ 0 & 1\end{array}\right)$ $\mbox{ (5 marks)}$

8.Given that $A=\left(\begin{array}{cc}7 & -1 \\ -2 & 1\end{array}\right), B=\left(\begin{array}{ll}7 & 6 \\ 6 & 5\end{array}\right)$ and $C=\left(\begin{array}{ll}3 & 2 \\ 1 & 4\end{array}\right)$, find $A^{-1}$ and $B^{-1}$, and use it to find the matrix $X$ such that $X B=C$. $\mbox{ (5 marks)}$

9.Let $A B=\left(\begin{array}{cc}4 & 2 \\ -5 & -3\end{array}\right)+I$ where $B=\left(\begin{array}{ll}1 & 2 \\ 4 & 9\end{array}\right) .$ Find $2 \times 2$ matrix $A$. $\mbox{ (5 marks)}$

10.Let $B=\left(\begin{array}{cc}4 & 2 \\ -5 & -3\end{array}\right)$.Find $B^{-1}$.Investigate whether or not squares of $B$ and $B^{-1}$ are inverse of each other. $\mbox{ (5 marks)}$

11.Find the solution set of the system of equations$$\begin{aligned}&2 x-3 y=19 \\&7 x+2 y=4\end{aligned}$$ by matrix method. $\mbox{ (5 marks)}$

12.Find the solution set of the following system of equations by matrix method : $$\begin{aligned}&7 x+4 y=16 \\&3 x+2 y=6\end{aligned}$$ $\mbox{ (5 marks)}$

13.Find the solution set of the system of equations by matrix method.$$\begin{array}{r}-2 x-y=0 \\x-2 y=5\end{array}$$ $\mbox{ (5 marks)}$

14.Solve the following system of equations by matrix method $$\begin{gathered}2 x+3 y=5 \\x-2 y=-1\end{gathered}$$ $\mbox{ (5 marks)}$

15.Find the solution set of the system of equations by matrix method: $$5 x+6 y=25,3 x+4 y=17$$ $\mbox{ (5 marks)}$

16.Find the solution set of the system of equations by matrix method: $$5 x-3 y=7,3 x+y=7$$ $\mbox{ (5 marks)}$

17.Find the solution set of the system of equations: $5 x-6 y+7=0, x+y-3=0$, by using matrix method. $\mbox{ (5 marks)}$

18.Find the solution set of the system of equations by matrix method.$$\begin{aligned}&3 x-2 y=0 \\&4 y+x=14\end{aligned}$$ $\mbox{ (5 marks)}$

19.Find the sclution set of the system $2 x-5 y=8$ $$3 y-7 x=1$$ by matrix method. $\mbox{ (5 marks)}$

20.Find the solution set of the system of equations by matrix method.$$\begin{aligned}&x-2 y=0 \\&4 y+x=9\end{aligned}$$ $\mbox{ (5 marks)}$

21.Use the matrix method to find the solution set of the simultaneous equations $5 x-2 y-11=0$ and $6 y+3 x-3=0$ $\mbox{ (5 marks)}$



Answer (2011)


$\quad\;\,$$\,$
1.$k=-4$
2.$k=3$
3.$m=4$
4.$\left(\begin{array}{lr}1 & 1 \\ 0 & -1\end{array}\right),\left(\begin{array}{cc}-1 & 1 \\ 0 & 1\end{array}\right)$
5.$a=2$
6.$k=1,\left(\begin{array}{lc}-2 & 1 \\ \frac{3}{2} & -\frac{1}{2}\end{array}\right)$
7.$\left(\begin{array}{ll}-1 & 1 \\ 0 & 1\end{array}\right)$
8.$A^{-1}=\left(\begin{array}{ll}\frac 15 & \frac 15 \\ \frac 25 & \frac 75\end{array}\right), B^{-1}=\left(\begin{array}{cc}-5 & 6 \\ 6 & -7\end{array}\right), X=\left(\begin{array}{cr}-3 & 4 \\ 19 & -22\end{array}\right)$
9.$A=\left(\begin{array}{lr}37 & -8 \\ -37 & 8\end{array}\right)$
10.$B^{-1}=\left(\begin{array}{ll}\frac 32 & 1 \\ -\frac 52 & -2\end{array}\right), B^{2}$ and $\left(B^{-1}\right)^{2}$ are inverses of each other.
11.$\{(2,-5)\}$
12.$\{(4,-3)\}$
13.$\{(1,-2)\}$
14.$x=1, y=1$
15.$\{(-1,5)\}$
16.$\{(2,1)\}$
17.$\{(1,2)\}$
18.$\{(2,3)\}$
19.$\{(-1,-2)\}$
20.$\left\{\left(3, \frac{3}{2}\right)\right\}$
21.$\left\{\left(2,-\frac{1}{2}\right)\right\}$


Group (2010)


$\quad\;\,$$\,$
1.If $A=\left(\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right)$ and $A^{2}=x A+y I$, find the values of $x$ and $y$.$\text{ (5 marks)}$

2.Given that $A=\left(\begin{array}{ll}3 & 2 \\ 4 & 4\end{array}\right)$ and $A^{2}+p A+4 I=O$, where $I$ is the unit matrix of order 2 , find the value of $p$.$\text{ (5 marks)}$

3.Given that $A=\left(\begin{array}{ll}3 & 3 \\ 4 & 3\end{array}\right)$ and $A^{2}+k A-3 I=O$, where $I$ is the unit matrix of order 2 , find the value of $k$.$\text{ (5 marks)}$

4.Given that $A=\left(\begin{array}{cc}5 & 1 \\ a+1 & a\end{array}\right)$ and $\operatorname{det} A=7$, find the value of $a$.If $I$ is the unit matrix of order 2, verify that $A^{2}-7 A+7 I=0$.$\text{ (5 marks)}$

5.If $A=\left(\begin{array}{ll}5 & 7 \\ 4 & 5\end{array}\right)$ and $A-3 A^{-1}-k I=O$, where $I$ is the unit matrix of order 2 , find $k$.$\text{ (5 marks)}$

6.Given that $A=\left(\begin{array}{ll}4 & 1 \\ 7 & 2\end{array}\right)$ and $B=\left(\begin{array}{cc}-2 & 3 \\ 6 & 8\end{array}\right)$, find $3 A^{\prime}+B A^{-1}$.$\text{ (5 marks)}$

7.Given that $A=\left(\begin{array}{ll}3 & 1 \\ 2 & 1\end{array}\right)$ and $B=\left(\begin{array}{ll}1 & 4 \\ 3 & 2\end{array}\right)$, find $\left(A^{-1}+B\right)(2 A-B)$.$\text{ (5 marks)}$

8.Given that $A=\left(\begin{array}{cc}4 & 3 \\ 1 & 1\end{array}\right)$ and $B=\left(\begin{array}{cc}4 & 2 \\ -5 & -3\end{array}\right)$ write down the inverse matrix $A^{-1}$ and use it to find the matrix $P$ and $Q$ such that $P A=2 I$ and $A Q=2 B$.$\text{ (5 marks)}$

9.Given that $A=\left(\begin{array}{cc}4 & 3 \\ 1 & 1\end{array}\right)$ and $B=\left(\begin{array}{cc}4 & 2 \\ -5 & -3\end{array}\right)$ write down the inverse matrix $B^{-1}$ and use it to find the matrix $P$ and $Q$ such that $P B=2 A$ and $B Q=4 A$.$\text{ (5 marks)}$

10.Given that $A=\left(\begin{array}{ll}3 & 2 \\ 1 & 4\end{array}\right)$ and $B=\left(\begin{array}{rr}k & 2 k \\ 3 & 4\end{array}\right)$, find the value of $k$ such that $\operatorname{det}(A B) \doteq-20$ and hence, find the inverse matrix of $B$.$\text{ (5 marks)}$

11.Find the $2 \times 2$ matrix $X$ in the equation: $$X\left(\begin{array}{cc}2 & -4 \\1 & 3\end{array}\right)=3\left(\begin{array}{cc}2 & -2 \\3 & 5\end{array}\right)-2\left(\begin{array}{cc}5 & -4 \\3 & 7\end{array}\right)$$ $\text{ (5 marks)}$

12.Find the solution set of the system of equations $$\begin{aligned}&4 x+3 y=24 \\&3 x+2 y=9\end{aligned}$$ by matrix method.$\text{ (5 marks)}$

13.Solve the following system of equations by matrix method.$$\begin{gathered}2 x+3 y=5 \\x-2 y=-1\end{gathered}$$ $\text{ (5 marks)}$

14.Use the matrix method to find the solution set of the simultaneous equations: $$\begin{aligned}&3 x-2 y=5 \\&2 x+3 y=12\end{aligned}$$ $\text{ (5 marks)}$

15.Find the solution set of the systems of equation $$\begin{aligned}&3 x+2 y=7 \\&5 x-y=3\end{aligned}$$ by matrix method.$\text{ (5 marks)}$

16.By using a matrix method, find the solution set of the system of equations $$\begin{aligned}&3 x+2 y=5 \\&5 x-3 y=21\end{aligned}$$ $\text{ (5 marks)}$

17.Find the solution set of the system of equations by matrix method: $$\begin{array}{r}3 x+y=10 \\y-x=-2\end{array}$$ $\text{ (5 marks)}$

18.Find the solution set of the system of equations by matrix method : $$\begin{aligned}&x+y=3 \\&y-x=1\end{aligned}$$ $\text{ (5 marks)}$

19.Find the solution'set of the equations : $$\begin{aligned}2 x-5 y &=-1 \\y-3 x &=-5, \text { by the matrix method.}\end{aligned}$$ $\text{ (5 marks)}$

20.Use the matrix method to find the solution set of the simultaneous equations : $$\begin{aligned}3 y-2 x &=1 \\x+2 y &=10\end{aligned}$$ $\text{ (5 marks)}$

21.Find the inverse of the matrix $\left(\begin{array}{ll}4 & 3 \\ 7 & 6\end{array}\right)$, and use it to solve the system of equations $$\begin{aligned}&7 x+6 y+16=0 \\&3 y+4 x+7=0\end{aligned}$$ $\text{ (5 marks)}$



Answer (2010)


$\quad\;\,$$\,$
1.$x=5, y=-7$
2.$-7$
3.$-6$
4.2
5.10
6.$\left(\begin{array}{ll}-13 & 35 \\ -41 & 32\end{array}\right)$
7.$\left(\begin{array}{ll}13 & -4 \\ 10 & -2\end{array}\right)$
8.$A^{-1}=\left(\begin{array}{cc}1 & -3 \\ -1 & 4\end{array}\right), P=\left(\begin{array}{lr}2 & -6 \\ -2 & 8\end{array}\right), Q=\left(\begin{array}{lr}38 & 22 \\ -48 & -28\end{array}\right)$
9.$B^{-1}=\left(\begin{array}{ll}\frac 32 & 1 \\-\frac 52 & 2\end{array}\right), P=\left(\begin{array}{ll}-3 & -4 \\ -2 & -2\end{array}\right), Q=\left(\begin{array}{lr}28 & 22 \\ -48 & -38\end{array}\right)$
10.$k=1, B^{-1}=\left(\begin{array}{cc}-2 & 1 \\ \frac 32 & -\frac 12\end{array}\right)$
11.$\left(\begin{array}{lc}-\frac 75 & -\frac 65\\ \frac 45 & \frac 75\end{array}\right)$
12.$\{(-21,36)\}$
13.$x=1, y=1$
14.$\{(3, 2)\} $
15.$\{(1,2)\}$
16.$\{(3,-2)\}$
17.$\{(3,1)\}$
18.$\{(1,2)\}$
19.$\{(2,1)\} $
20.$\{(4,3)\}$
21.$\left(\begin{array}{ll}2 & -1 \\ -\frac 73 & \frac 43\end{array}\right) ; x=2, y=-5$

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