# CIE Matrices (Additional Mathematics -2018)

$\def\D{\displaystyle}\def\frac{\dfrac}$
$\newcommand{\iixii}{\left(\begin{array}{cc}#1&#2\\#3&#4\end{array}\right)}$
$\newcommand{\matrixa}{\left(\begin{array}{cc}#1\end{array}\right)}$

1 (CIE 2012, s, paper 21, question 1)
(i) Given that $\D A= \iixii{4}{-3}{2}{5},$ find the inverse matrix $\D A^{-1}.$ 
(ii) Use your answer to part (i) to solve the simultaneous equations
$\D 4x - 3y = -10,$
$\D 2x + 5y = 21. $

2 (CIE 2012, s, paper 22, question 4)
In a competition the contestants search for hidden targets which are classed as difficult, medium or easy. In the first round, finding a difficult target scores 5 points, a medium target 3 points and an easy target 1 point. The number of targets found by the two contestants, Claire and Denise, are shown in the table.
$\begin{array}{|l|c|c|c|}\hline \text { Contestant } \backslash \text { Target } & \text { Difficult } & \text { Medium } & \text { Easy } \\\hline \text { Claire } &4 & 1 & 7 \\\hline \text { Denise } &2 & 5 & 1 \\\hline\end{array}$
In the second round, finding a difficult target scores 8 points, a medium target 4 points and an easy target 2 points. In the second round Claire finds 2 difficult, 5 medium and 2 easy targets whilst Denise finds 4 difficult, 3 medium and 6 easy targets.
(i) Write down the sum of two matrix products which, on evaluation, would give the total score for each contestant. 
(ii) Use matrix multiplication and addition to calculate the total score for each contestant. 

3 (CIE 2012, w, paper 12, question 2)
(i) Find the inverse of the matrix $\D \iixii{2}{-1}{-1}{1.5}.$ 
(ii) Hence find the matrix A such that $\D \iixii{2}{-1}{-1}{1.5}A=\iixii{1}{6}{-0.5}{4}.$ 

4 (CIE 2012, w, paper 21, question 5)
It is given that $\D A=\matrixa{ 4& -2\\8& -3}, B = \matrixa{2& 0& 4\\5& -1& 4}$ and $\D C = \matrixa{5\\-2\\3}.$
(i) Calculate $\D ABC.$ 
(ii) Calculate $\D A^{-1} B.$ 

5 (CIE 2012, w, paper 23, question 2)
(i) Given that $\D A = \iixii{7}{8}{4}{6}$, find the inverse matrix, $\D A^{-1}.$ 
(ii) Use your answer to part (i) to solve the simultaneous equations
$\D 7x + 8y = 39,$
$\D 4x + 6y = 23.$ 

6 (CIE 2013, s, paper 11, question 6)
(i) Given that $\D A=\iixii{2}{-1}{3}{5}$, find $\D A^{-1}.$ 
(ii) Using your answer from part (i), or otherwise, find the values of $\D a, b, c$ and $\D d$ such that
$A\iixii{a}{b}{c}{-1}=\iixii{7}{5}{17}{d}.$ 

7 (CIE 2013, s, paper 12, question 8)
(a) Given that the matrix $\D A =\iixii{4}{2}{3}{-5},$ find
(i) $\D A^2,$ 
(ii) $\D 3A + 4I,$ where $\D I$ is the identity matrix. 
(b) Find the inverse matrix of $\D \iixii{6}{1}{-9}{3}.$
Hence solve the equations
$\D 6x+y=5$,
$\D -9x+3y=\frac{3}{2}.$

8 (CIE 2013, w, paper 11, question 11)
(a) It is given that the matrix$\D A=\iixii{2}{3}{4}{1}.$
(i) Find $\D A + 2I.$ 
(ii) Find $\D A^2.$ 
(iii) Using your answer to part (ii) find the matrix $\D B$ such that $\D A^2B = I.$ 
(b) Given that the matrix $\D C=\iixii{x}{-1}{x^2-x+1}{x-1},$ show that $\D \det C\not=0.$ 

9 (CIE 2013, w, paper 13, question 7)
It is given that $\D A=\iixii{2t}{2}{t^2-t+1}{t}.$
(i) Find the value of $\D t$ for which $\D \det A = 1.$ 
(ii) In the case when $\D t = 3,$ find $\D A^{-1}$ and hence solve
$\D 3x + y = 5,$
$\D 7x + 3y = 11.$ 

10 (CIE 2014, s, paper 11, question 6)
Matrices $\D A$ and $\D B$ are such that $\D A =\matrixa{-1&4\\7&6\\4&2}.$ and $\D B =\iixii{2}{1}{3}{5}.$
(i) Find $\D AB.$ 
(ii) Find $\D B^{-1} .$ 
$\D 4x+2y=-3$,
$\D 6x+10y=-22.$ 

11 (CIE 2014, s, paper 12, question 6)
(a) Matrices $\D X, Y$ and $\D Z$ are such that $\D X=\iixii{2}{3}{1}{2}, Y=\matrixa{1&3\\4&5\\6&7}$ and $\D Z=\matrixa{1&2&3}.$ Write down all the matrix products which are possible using any two of these matrices. Do not evaluate these products. 
(b) Matrices $\D A$ and $\D B$ are such that $\D A=\iixii{5}{-2}{-4}{1}$ and $\D AB= \iixii{3}{9}{-6}{-3}.$ Find the matrix $\D B.$ 

12 (CIE 2014, s, paper 23, question 3)
In a motor racing competition, the winning driver in each race scores 5 points, the second and third placed drivers score 3 and 1 points respectively. Each team has two members. The results of the drivers in one team, over a number of races, are shown in the table below.
$\begin{array}{lccc}\text{Driver}& \text{1st place }& \text{2nd place}& \text{3rd place}\\\text{Alan }& 3 & 1 & 4\\\text{Brian }& 1 & 4 & 0\end{array}$
(i) Write down two matrices whose product under matrix multiplication will give the number of points scored by each of the drivers. Hence calculate the number of points scored by Alan and by Brian. 
(ii) The points scored by Alan and by Brian are added to give the number of points scored by the team. Using your answer to part (i), write down two matrices whose product would give the number of points scored by the team. 

13 (CIE 2014, w, paper 11, question 7)
Matrices $\D A$ and $\D B$ are such that $\D \iixii{3a}{2b}{-a}{b}$ and $\D B=\iixii{-a}{b}{2a}{2b},$ where $\D a$ and $\D b$ are non-zero constants.
(i) Find $\D A^{-1}.$  
(ii) Using your answer to part (i), find the matrix $\D X$ such that $\D XA = B.$ 

14 (CIE 2014, w, paper 13, question 5)
(a) A drinks machine sells coffee, tea and cola. Coffee costs \$0.50, tea costs \$0.40 and cola costs
\$0.45. The table below shows the numbers of drinks sold over a 4-day period.$\begin{array}{lccc}                        & \text{Coffee}& \text{Tea}& \text{Cola}\\\text{Tuesday }&             12 &      2 &    1\\\text{Wednesday }&         9   &     3 &    0\\\text{Thursday}&             8 &       5 &    1\\\text{Friday  }&               11 &      2&     0\end{array}$(i) Write down 2 matrices whose product will give the amount of money the drinks machine took each day and evaluate this product.  (ii) Hence write down the total amount of money taken by the machine for this 4-day period.  (b) Matrices$\D X$and$\D Y$are such that$\D X = \iixii{2}{4}{-5}{1}$and$\D XY = I,$where$\D I$is the identity matrix. Find the matrix$\D Y.$ 15 (CIE 2015, s, paper 11, question 4) (a) Given that the matrix$\mathbf{X}=\left(\begin{array}{rr}2 & -4 \\ k & 0\end{array}\right)$, find$\mathbf{X}^{2}$in terms of the constant$k$.$$(b) Given that the matrix$\mathrm{A}=\left(\begin{array}{ll}a & 1 \\ b & 5\end{array}\right)$and the matrix$\mathrm{A}^{-1}=\left(\begin{array}{rr}\frac{5}{6} & -\frac{1}{6} \\ -\frac{2}{3} & \frac{1}{3}\end{array}\right)$, find the value of each of the integers$a$and$b$. 16 CIE$2015, \mathrm{~s}$, paper 12, question 3$)$Find the inverse of the matrix$\left(\begin{array}{ll}4 & 2 \\ 5 & 3\end{array}\right)and hence solve the simultaneous equations \begin{aligned}&4 x+2 y-8=0 \\&5 x+3 y-9=0\end{aligned} 17 (CIE 2015, s, paper 21, question 3) (a) Find the matrix\mathrm{A}$if$\quad 4 \mathrm{~A}+5\left(\begin{array}{rrr}4 & 0 & -1 \\ 3 & -2 & 5\end{array}\right)=\left(\begin{array}{rrr}52 & -8 & 19 \\ 31 & 2 & 65\end{array}\right)$.$$(b)$\mathbf{P}=\left(\begin{array}{ccc}30 & 25 & 65 \\ 70 & 15 & 80 \\ 50 & 40 & 30 \\ 40 & 20 & 75\end{array}\right) \quad \mathbf{Q}=\left(\begin{array}{llll}650 & 500 & 450 & 225\end{array}\right)$The matrix$\mathbf{P}$represents the number of 4 different televisions that are on sale in each of 3 shops. The matrix$\mathbf{Q}$represents the value of each television in dollars. (i) State, without evaluation, what is represented by the matrix QP.$$(ii) Given that the matrix$\mathbf{R}=\left(\begin{array}{l}1 \\ 1 \end{array}\right)$, state, without evaluation, what is represented by the matrix$QPR$. 1 18 (CIE 2015, s, paper 22, question 2) The table shows the number of passengers in Economy class and in Business class on 3 flights from London to Paris, The table also shows the departure times for the 3 flights and the cost of a single ticket in each class.$\begin{array}{|c|c|c|}\hline \text{Departure time} & \text{Number of passengers}  & \text{Number of passengers}  \\ &\text{in Economy class}&\text{in Business class}\\\hline 0930 & 60 & 50 \\\hline 1330 & 70 & 52 \\\hline 1545 & 58 & 34 \\\hline \hline \text{Single ticket price (pound)} & 120 & 300 \\\hline\end{array}$(i) Write down a matrix,$\mathbf{P}$, for the numbers of passengers and a matrix,$\mathbf{Q}$, of single ticket prices, such that the matrix product QP can be found. (ii) Find the matrix product QP.$$(iii) Given that$\mathbf{R}=\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)$, explain what information is found by evaluating the matrix product QPR.  19 (IE$2015, \mathrm{w}$, paper 13 , question 3) (a) Matrices$\mathbf{A}$and$\mathbf{B}$are such that$\mathbf{A}=\left(\begin{array}{ll}2 & 1 \\ 4 & 3\end{array}\right)$and$\mathbf{B}=\left(\begin{array}{lll}3 & 8 & 1 \\ 6 & 0 & 2\end{array}\right) .$Find$\mathbf{A B}$.$$(b) Given that matrix$\mathbf{X}=\left(\begin{array}{rr}4 & 6 \\ 2 & -8\end{array}\right)$, find the integer value of$m$and of$n$such that$\mathbf{X}^{2}=m \mathbf{X}+n \mathbf{I}$where$\mathbf{I}$is the identity matrix. (c) Given that matrix$\mathbf{Y}=\left(\begin{array}{ll}a & 2 \\ 3 & a\end{array}\right)$, find the values of$a$for which det$\mathbf{Y}=0$.$$20 (CIE$2015, \mathrm{w}$, paper 21, question 4$)$(a) Given that$\mathbf{A}=\left(\begin{array}{rr}2 & -1 \\ 3 & 5 \\ 7 & 4\end{array}\right)$and$\mathbf{B}=\left(\begin{array}{rrr}1 & -2 & 4 \\ -2 & 3 & 0\end{array}\right)$, calculate$2 \mathbf{B A}$.$$(b) The matrices$\mathbf{C}$and$\mathbf{D}$are given by$\mathbf{C}=\left(\begin{array}{rr}1 & 2 \\ -1 & 6\end{array}\right)$and$\mathbf{D}=\left(\begin{array}{rr}3 & -2 \\ 1 & 4\end{array}\right)$. (i) Find$\mathrm{C}^{-1}$. (ii) Hence find the matrix$\mathbf{X}$such that$\mathbf{C X}+\mathbf{D}=\mathbf{I}$, where$\mathbf{I}$is the identity matrix.$$21 (CIE 2016, march, paper 22, question 7) (a) Given that$\mathbf{A}=\left(\begin{array}{rrr}4 & 6 & 8 \\ -2 & 0 & 4\end{array}\right)$and$\mathbf{B}=\left(\begin{array}{rrr}6 & 1 & 2 \\ 7 & -2 & 1\end{array}\right)$, find$\mathbf{A}-3 \mathbf{B}$. (b) Given that$\mathbf{C}=\left(\begin{array}{rr}-2 & 0 \\ 4 & 1\end{array}\right)$and$\mathbf{D}=\left(\begin{array}{rr}4 & 3 \\ -3 & -5\end{array}\right)$, find (i) the inverse matrix$\mathbf{C}^{-1}$, (ii) the matrix$\mathbf{X}$such that$\mathbf{X D}^{-1}=\mathbf{C}$. 22 CIE 2016, s, paper 11, question 4) (a) Given the matrices$\mathbf{A}=\left(\begin{array}{rr}-1 & 2 \\ 3 & 0\end{array}\right)$and$\mathbf{B}=\left(\begin{array}{ll}3 & 0 \\ 1 & 2\end{array}\right)$, find$\mathbf{A}^{2}-2 \mathbf{B}$.$$(b) Using a matrix method, solve the equations $$\begin{array}{r}4 x+y=1 \\10 x+3 y=1\end{array}$$ 23 (CIE 2016, s, paper 22, question 7) The matrix$\mathbf{A}$is$\left(\begin{array}{ll}4 & 5 \\ 3 & 2\end{array}\right)$and the matrix$\mathbf{B}$is$\left(\begin{array}{ll}4 & 2 \\ 1 & 3\end{array}\right)$. (i) Find the matrix$\mathbf{C}$such that$\mathbf{C}=3 \mathbf{A}+\mathbf{B}$. (ii) Show that$\operatorname{det}(\mathbf{A B})=\operatorname{det} \mathbf{A} \times \operatorname{det} \mathbf{B}$. (iii) Find the matrix$(\mathbf{A B})^{-1}$. 24 (CIE 2016, w, paper 11, question 6) (a) Matrices$\mathbf{X}, \mathbf{Y}$and$\mathbf{Z}$are such that $$\mathbf{X}=\left(\begin{array}{rr}2 & 3 \\4 & -1 \\6 & 5\end{array}\right), \mathbf{Y}=\left(\begin{array}{lll}1 & -1 & 0\end{array}\right) \text { and } \mathbf{Z}=\left(\begin{array}{rr}0 & -1 \\5 & 3\end{array}\right) .$$ Write down all the matrix products which are possible using any two of these matrices. Do not evaluate these products. (b) Matrices$\mathrm{A}, \mathbf{B}$and$\mathbf{C}$are such that$\mathbf{A}=\left(\begin{array}{rr}2 & -1 \\ 4 & 7\end{array}\right), \mathbf{B}=\left(\begin{array}{rr}-4 & 2 \\ 10 & 4\end{array}\right)$and$\mathbf{A C}=\mathbf{B}$. (i) Find$\mathrm{A}^{-1}$. (ii) Hence find$\mathbf{C}$. 25 (CIE$2016, \mathrm{w}$, paper 23 , question 11) It is given that$\mathbf{A}=\left(\begin{array}{ll}2 & q \\ p & 3\end{array}\right)$and that$\mathbf{A}^{2}-5 \mathbf{A}=2 \mathbf{1}$, where$\mathbf{I}$is the identity matrix. (i) Find a relationship connecting the constants$p$and$q$. (ii) Given that$p$and$q$are positive and that det$\mathbf{A}=-3 p$, find the value of$p$and of$q$. 26 CIE 2017, march, paper 12, question 4) (a) It is given that$\mathbf{A}=\left(\begin{array}{rr}1 & -3 \\ 2 & 4\end{array}\right)$(i) Find$\mathbf{A}^{-1}$. (ii) Using your answer to part (i), find the matrix$\mathbf{M}$such that$\mathbf{A M}=\left(\begin{array}{rr}-1 & -5 \\ 4 & 2\end{array}\right)$. (b)$\mathbf{X}$is$\left(\begin{array}{ll}a & -1 \\ 2 & -3\end{array}\right)$and$\mathbf{Y}$is$\left(\begin{array}{cc}2 & 1 \\ 4 & 3 a\end{array}\right)$, where$a$is a constant. Given that$\operatorname{det} \mathbf{X}=4$det$\mathbf{Y}$, find the value of$a$27 (CIE 2017, s, paper 12, question 9) (a) Given that$\mathbf{A}=\left(\begin{array}{rr}3 & 1 \\ -1 & 2 \\ 4 & 5\end{array}\right), \mathbf{B}=\left(\begin{array}{rr}1 & -2 \\ 3 & 0\end{array}\right)$and$\mathbf{C}=\mathbf{A B}$, (i) state the order of$\mathbf{A}$, (ii) find C.$$(b) The matrix$\mathbf{X}=\left(\begin{array}{rr}5 & -12 \\ 4 & -7\end{array}\right)$. (i) Find$\mathbf{X}^{-1}$. (ii) Using$\mathbf{X}^{-1}$, find the coordinates of the point of intersection of the lines$12 y=5 x-267 y=4 x-52 .$28 (CIE 2017, s, paper 13, question 6) (a) Given that$\mathbf{A}=\left(\begin{array}{ll}3 & 1 \\ 2 & 4\end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rr}5 & 1 \\ 2 & 4 \\ -1 & 0\end{array}\right)$and$\mathbf{C}=\left(\begin{array}{rr}-5 & 2 \\ 3 & 1\end{array}\right)$, find (i)$\mathrm{A}+3 \mathrm{C}$,$$(ii)$\mathrm{BA}$.$$(b) (i) Given that$\mathbf{X}=\left(\begin{array}{ll}1 & -3 \\ 4 & -2\end{array}\right)$, find$\mathbf{X}^{-1}$.$$(ii) Hence find$\mathbf{Y}$, such that$\mathbf{X Y}=\left(\begin{array}{rr}5 & -10 \\ 15 & 20\end{array}\right)$. 29 CIE 2017, s, paper 21, question 6) Four cinemas,$P, Q, R$and$S$each sell adult, student and child tickets. The number of tickets sold by each cinema on one weekday were$P: 90$adult, 10 student, 30 child$Q: 45$student$R: 25$adult, 15 child$S: 10$adult, 100 child. (i) Given that$\mathbf{L}=\left(\begin{array}{llll}1 & 1 & 1 & \text { 1), construct a matrix, } \mathbf{M} \text {, of the number of tickets sold, such that the }\end{array}\right.$matrix product LM can be found. (ii) Find the matrix product$\mathbf{L M}$. (iii) State what information is represented by the matrix product LM. An adult ticket costs$\$5$, a student ticket costs $\$ 4$and a child ticket costs$\$3$.
(iv) Construct a matrix, $\mathbf{N}$, of the ticket costs, such that the matrix product LMN can be found and state what information is represented by the matrix product LMN.

30 (CIE 2017, w, paper 13, question 10)
(a) Given that $\mathbf{A}=\left(\begin{array}{rr}4 & -1 \\ a & b\end{array}\right), \mathbf{B}=\left(\begin{array}{rr}2 & 3 \\ -5 & 4\end{array}\right)$ and $\mathrm{A} \mathbf{B}=\left(\begin{array}{ll}13 & 8 \\ 18 & 4\end{array}\right)$, find the value of $a$ and of $b$.
(b) It is given that $\mathbf{X}=\left(\begin{array}{rr}3 & -5 \\ -4 & 1\end{array}\right), \mathbf{Y}=\left(\begin{array}{rr}-1 & 2 \\ 4 & 0\end{array}\right)$ and $\mathbf{X Z}=\mathbf{Y}$.
(i) Find $\mathbf{X}^{-1}$.
(ii) Hence find \mathbf{Z}$$ 31 (CIE 2017, w, paper 21, question 6) It is given that \mathbf{M}=\left(\begin{array}{rr}2 & p \\ -3 & q\end{array}\right) where p and q are integers. (i) If det \mathbf{M}=13, find an equation connecting p and q. (ii) Given also that \mathbf{M}^{2}=\left(\begin{array}{rc}4-3 p & 12 \\ -6-3 q & -3 p+q^{2}\end{array}\right), find a second equation connecting p and q. (iii) Find the value of p and of q. 32 CIE 2017, w, paper 22, question 8) The matrix \mathbf{A} is \left(\begin{array}{ll}2 & 1 \\ 4 & 3\end{array}\right). (i) Find (2 \mathbf{A})^{-1}. (ii) Hence solve the simultaneous equations$$\begin{aligned}&2 y+4 x+5=0 \\&6 y+8 x+9=0\end{aligned}$33 (CIE 2018, march, paper 12, question 7) (a) Find the values of$a$for which$\operatorname{det}\left(\begin{array}{rr}2 a & 1 \\ 4 a & a\end{array}\right)=6-3 a$.$$(b) It is given that$\mathbf{A}=\left(\begin{array}{ll}2 & 1 \\ 3 & 4\end{array}\right)$and$\mathbf{B}=\left(\begin{array}{rr}2 & 0 \\ -3 & 5\end{array}\right)$. (i) Find$\mathbf{A}^{-1}$.  (ii) Hence find the matrix$\mathbf{C}$such that$\mathbf{A C}=\mathbf{B}$. (c) Find the$2 \times 2$matrix$\mathbf{D}$such that$4 \mathbf{D}+3 \mathbf{I}=\mathbf{O}$. 34 (CIE 2018, s, paper 11, question 7) (i) Find the inverse of the matrix$\left(\begin{array}{rr}4 & -2 \\ -5 & 3\end{array}\right)$. (ii) Hence solve the simultaneous equations$8 x-4 y-5=0$,$-10 x+6 y-7=0$35 (CIE 2018, s, paper 22, question 8 (a)$\mathbf{A}=\left(\begin{array}{cc}2 & -1 \\ 1 & -3\end{array}\right)$and$\mathbf{B}=\left(\begin{array}{ll}0 & -2 \\ 3 & -5\end{array}\right)$. Find$(\mathbf{B A})^{-1}$. (b) The matrix$\mathbf{X}$is such that$\mathbf{X C}=\mathbf{D}$, where$\mathbf{C}=\left(\begin{array}{rrr}-2 & 5 & 3 \\ 0 & 10 & 4\end{array}\right)$and$\mathbf{D}=\left(\begin{array}{lll}-4 & 5 & 4\end{array}\right)$. (i) State the order of the matrix$\mathbf{C}$. (ii) Find the matrix$\mathbf{X}$.$$Answers$\def\frac{\dfrac}$1.(i)$\D \frac{1}{26}\iixii{5}{3}{-2}{4}$(ii)$\D x=0.5,y=4$2(i)$\D \matrixa{4&1&7\\2&5&1} \matrixa{5\\3\\1} +\matrixa{2&5&2\\4&3&6}\matrixa{8\\4\\1}$(ii) Claire=70, Denise=82 3(i)$\D \frac{1}{2}\matrixa{1.5&1\\1&2}$(ii)$\D \matrixa{.5&6.5\\0&7}$4(i)$\D \matrixa{10\\59}$(ii)$\D \matrixa{1&-0.5&-1\\1&-1&-4}$5(i)$\D \frac{1}{10}\matrixa{6&-8\\-4&7}$(ii)$\D x=5,y=0.5$6(i)$\D A^{-1}\dfrac{1}{13}\matrixa{5&1\\-3&2}$(ii)$ \D a=4,b=2,c=-1,d=1 $7 (a)(i)$\D \matrixa{22&-2\\-3&31}$(ii)$\D \matrixa{16&6\\9&-11}$(b)(i)$\D \dfrac{1}{27}\matrixa{3&-1\\9&6}$(ii)$\D x=0.5,y=2$8(a)(i)$\D \matrixa{4&3\\4&3}$(ii)$\D A^2=\matrixa{16&9\\12&13}$(iii)$\D \dfrac{1}{100}\matrixa{13&-9\\-12&16}$9(i)$\D t=\dfrac{3}{2}$(ii)$\D A=\matrixa{6&2\\7&3}, A^{-1}=\dfrac{1}{4}\matrixa{3&-2\\-7&6}\D x=2,y=-1$10 (i)$\D \matrixa{10&19\\32&37\\14&14}$(ii)$\D B^{-1}=\dfrac{1}{7}\matrixa{5&-1\\-3&2}\D   x=0.5,y=-2.5$11(a)$\D YX,ZY$(b)$\D \matrixa{3&-1\\6&-7}$12(i)$\D \matrixa{3&1&4\\1&3&0},\matrixa{5\\3\\1}, \matrixa{22\\17}$(ii)$\D (1,1),\matrixa{22\\17}$13(i)$\D A^{-1} =\dfrac{1}{5ab}\matrixa{b&-2b\\ a&3a}$(ii)$\D X=\matrixa{0&1\\4/5&2/5}$14(a)(i)$\D \matrixa{0.5&0.4&0.45} \matrixa{12&9&8&11\\ 2&3&5&2\\ 1&0&1&0} =\matrixa{7.25&5.70&6.45&6.30}$(ii) 25.70 (b)$\D \dfrac{1}{22}\matrixa{1&-4\\5&2}$15. (a)$\left(\begin{array}{cc}4-4 k & -8 \\ 2 k & -4 k\end{array}\right)$(b)$a=2, b=4$16.$A^{-1}=\frac{1}{2}\left(\begin{array}{cc}3 & -2 \\ -5 & 4\end{array}\right)x=3, y=-2$17. (a)$A=\left(\begin{array}{ccc}8 & -2 & 6 \\ 4 & 3 & 10\end{array}\right)$(b)(i) The (total) value of the stock in each of the 3 shops (ii) The total value of the stock in all 3 shops 18. (i)$P=\left(\begin{array}{lll}60 & 70 & 58 \\ 50 & 52 & 34\end{array}\right)Q=\left(\begin{array}{ll}120 & 300\end{array}\right)$(ii)$\left(\begin{array}{lll}22200 & 24000 & 17160\end{array}\right)$(iii) The total from all flights 19. (a)$\left(\begin{array}{ccc}12 & 16 & 4 \\ 30 & 32 & 10\end{array}\right)$(b)$n=44$(c)$a=\pm \sqrt{6}$20. (a)$\left(\begin{array}{ll}48 & 10 \\ 10 & 34\end{array}\right)$(b)(i)$\frac{1}{8}\left(\begin{array}{cc}6 & -2 \\ 1 & 1\end{array}\right)$(ii)$\frac{1}{8}\left(\begin{array}{cc}-10 & 18 \\ -3 & -1\end{array}\right)$21. (a)$\left(\begin{array}{lll}-14 & 3 & 2 \\ -23 & 6 & 1\end{array}\right)$(b)(i)$-\frac 12\left(\begin{array}{cc}1 & 0 \\ -4 & -2\end{array}\right)$(b)(ii)$\left(\begin{array}{cc}-8 & -6 \\ 13 & 7\end{array}\right)$22. (a)$\left(\begin{array}{cc}1 & -2 \\ -5 & 2\end{array}\right)$(b)$x=1, y=-3$23. (i)$\left(\begin{array}{cc}16 & 17 \\ 10 & 9\end{array}\right)$(iii)$\frac{1}{-70}\left(\begin{array}{cc}12 & -23 \\ -14 & 21\end{array}\right)$24. (a) YX, XZ (b)(i)$\frac{1}{18}\left(\begin{array}{cc}7 & 1 \\ -4 & 2\end{array}\right)$(ii)$\left(\begin{array}{cc}-1 & 1 \\ 2 & 0\end{array}\right)$25. (i)$p q=8($ii$) p=2 / 3, q=12$26. (a)(i)$\frac{1}{10}\left(\begin{array}{cc}4 & 3 \\ -2 & 1\end{array}\right)$(ii)$\frac{1}{5}\left(\begin{array}{cc}4 & -7 \\ 3 & 6\end{array}\right)$(b)$\frac 23$27. (a)(i)$3 \times 2$(a)(ii)$\left(\begin{array}{cc}6 & -6 \\ 5 & 2 \\ 19 & -8\end{array}\right)$(b)(i)$X^{-1}=\frac{1}{13}\left(\begin{array}{cc}-7 & 12 \\ -4 & 5\end{array}\right)$(b)(ii)$x=34, y=12$28. (a)(i)$\left(\begin{array}{cc}-12 & 7 \\ 11 & 7\end{array}\right)$(ii)$\left(\begin{array}{cc}17 & 9 \\ 14 & 18 \\ -3 & -1\end{array}\right)$(b)(i)$X^{-1}=\frac{1}{10}\left(\begin{array}{ll}-2 & 3 \\ -4 & 1\end{array}\right)$(b)(ii)$\left(\begin{array}{cc}3.5 & 8 \\ -0.5 & 6\end{array}\right)$29. (i)$\left(\begin{array}{ccc}90 & 10 & 30 \\ 0 & 50 & 0 \\ 25 & 0 & 15 \\ 10 & 0 & 100\end{array}\right)$(II)$\left(\begin{array}{lll}125 & 55 & 145\end{array}\right)$(iii) total of each type (iv)$\left(\begin{array}{l}5 \\ 4 \\ 3\end{array}\right)$30. (a)$a=4, b=-2$(b)(i)$-\frac{1}{17}\left(\begin{array}{ll}1 & 5 \\ 4 & 3\end{array}\right)$(b)(ii)$-\frac{1}{17}\left(\begin{array}{cc}19 & 2 \\ 8 & 8\end{array}\right)$31. (i)$2 p+3 q=13$(ii)$2 p+p q=12$(iii)$p=3, q=2$32. (i)$(1 / 8)\left(\begin{array}{cc}6 & -2 \\ -8 & 4\end{array}\right)$(ii)$x=-1.5, y=0.5$33. (a)$a=2,-3 / 2$(b)(i)$\frac 15\left(\begin{array}{cc}4 & -1 \\ -3 & 2\end{array}\right)$(ii)$\frac 15\left(\begin{array}{cc}11 & -5 \\ -12 & 10\end{array}\right)$(c)$\left(\begin{array}{cc}-3 / 4 & 0 \\ 0 & -3 / 4\end{array}\right)$34.$\frac{1}{2}\left(\begin{array}{ll}3 & 2 \\ 5 & 4\end{array}\right)x=7.25, y=13.25$35. (a)$\frac{1}{-30}\left(\begin{array}{cc}12 & -6 \\ -1 & -2\end{array}\right)$(b)(i)$2 \times 3$(b)(ii)$(2-\frac 12)\$