Higher Level Mathematics

Ibrahim Wazir, Tim Garry, Peter Ashbourne

Chapter 6

Matrix Algebra - all with Video Answers

Educators

Section 1

Basic definitions

Problem 1

Consider the following matrices
$$A=\left(\begin{array}{cc}-2 & x \\y-1 & 3\end{array}\right), B=\left(\begin{array}{cc}x+1 & -3 \\
4 & y-2\end{array}\right)$$
a) Evaluate each of the following
(i) $A+B$
(ii) $3 A-B$
b) Find $x$ and $y$ such that $A=B$
c) Find $x$ and $y$ such that $A+B$ is a diagonal matrix.
d) Find $A B$ and $B A$.

SS

Sarvesh Somasundaram

Numerade Educator

Problem 2

Solve for the variables.
a) $\left(\begin{array}{ll}3 & 0 \\ 4 & 2\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{r}6 \\ -12\end{array}\right)$
b) $\left(\begin{array}{ll}2 & p \\ 3 & q\end{array}\right)\left(\begin{array}{l}4 \\ 5\end{array}\right)=\left(\begin{array}{c}18 \\ -8\end{array}\right)$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 3

The diagram below shows the major highways connecting some European cities: Vienna ( $V$ ), Munich ( $M$ ), Frankfurt ( $F$ ), Stuttgart (S), Zurich (Z), Milano (L) and Paris ( $P$ ). (GRAPH CAN'T COPY)
a) Write the number of direct routes between each pair of cities into a matrix as started below:
(TABLE CAN'T COPY)
b) Multiply the matrix from part a) by itself and interpret what it signifies.

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 4

Consider the following matrices: $A=\left(\begin{array}{rrr}2 & 5 & 1 \\ 0 & -3 & 2 \\ 7 & 0 & -1\end{array}\right), B=\left(\begin{array}{rr}m & -2 \\ 3 m & -1 \\ 2 & 3\end{array}\right), C=\left(\begin{array}{ccc}x-1 & 5 & y \\ 0 & -x & y+1 \\ 2 x+y & x-3 y & 2 y-x\end{array}\right)$
a) Find $A+C$
b) Find $A B$
c) Find BA.
d) Solve for $x$ and $y$ if $A=C$
e) Find $B+C$
f) Solve for $m$ if $3 B+2\left(\begin{array}{cc}-1 & m^{2} \\ -5 & 2 \\ 1 & -1\end{array}\right)=\left(\begin{array}{cc}7 & 12 \\ 17 & 1 \\ 2 m+2 & 7\end{array}\right)$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 5

Find $a, b$ and $c$ so that the following equation is true:
$2 \cdot\left(\begin{array}{ll}a-1 & b \\ c+2 & 3\end{array}\right)+\left(\begin{array}{rr}3 & -1 \\ 0 & 5\end{array}\right)=\left(\begin{array}{cc}-5 & 5 \\ 8 & c+9\end{array}\right)$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 6

Find $x$ and $y$ such that:
$\left(\begin{array}{rr}2 & -3 \\ -5 & 7\end{array}\right)\left(\begin{array}{cc}x-11 & 1-x \\ -5 & x+2 y\end{array}\right)=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 7

Find $m$ and $n$ if
$\left(\begin{array}{cc}m^{2}-1 & m+2 \\ 5 & -2\end{array}\right)=\left(\begin{array}{cc}3 & n+1 \\ 5 & n-5\end{array}\right)$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 8

There are two supermarkets in your area. Your shopping list consists of $2 \mathrm{kg}$ of tomatoes, $500 \mathrm{g}$ of meat and 3 litres of milk. Prices differ between the different shops, and it is difficult to switch between stores to make certain you are paying the least amount of money. A better strategy is to check and see where you pay less on average! The prices of the different items are given below. Which shop should you go to?
$$\begin{array}{|l|c|c|}\hline \text { Product } & \text { Price in shop A } & \text { Price in shop B } \\
\hline \text { Tomato } & € 1.66 / \mathrm{kg} & € 1.58 / \mathrm{kg} \\\hline \text { Meat } & € 2.55 / 100 \mathrm{g} & € 2.6 / 100 \mathrm{g} \\\hline \text { Milk } & € 0.90 \text { /litre } & € 0.95 / \text { litre } \\\hline\end{array}$$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 9

Consider the matrices $A=\left(\begin{array}{rr}2 & 0 \\ -5 & 1\end{array}\right), B=\left(\begin{array}{rr}3 & -1 \\ 1 & 4\end{array}\right)$ and $C=\left(\begin{array}{rr}-3 & 5 \\ 2 & 7\end{array}\right)$
a) Find $A+(B+C)$ and $(A+B)+C$
b) Make a conjecture about the addition of $2 \times 2$ matrices observed in a) above and prove it.
c) Find $A(B C \text { and }(A B) C$.
d) Make a conjecture about the multiplication of $2 \times 2$ matrices observed in $c$ ) above and prove it.

Titus Dascalu

Numerade Educator

Problem 10

A company stores and sells air conditioning units, electric heaters and humidifiers. Row matrix A represents the number of each unit sold last year, and matrix $B$ represents the profit margin for each unit. Find $A B$ and describe what the product represents. $A=(235 \quad 562 \quad 117), B=\left(\begin{array}{c}\epsilon 120 \\ 695 \\ \epsilon 56\end{array}\right)$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 11

Find $r$ and $s$ such that the following equation is true: $r A+B=A$, where $A=\left(\begin{array}{ll}2 & 3 \\ 5 & 7\end{array}\right)$ and $B=\left(\begin{array}{cc}-4 & -6 \\ s-8 & -14\end{array}\right)$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 12

Let $A=\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right)$
a) Find:
(i) $A^{2}$
(ii) $A^{3}$
(iii) $A^{4}$
(iv) $A^{n}$
Let $B=\left(\begin{array}{ll}3 & 3 \\ 0 & 3\end{array}\right)$
b) Find:
(i) $B^{2}$
(ii) $B^{3}$
(iii) $B^{4}$
(iv) $B^{n}$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 13

Solve for $x$ and $y$ such that $A B=B A$ if $A=\left(\begin{array}{ll}2 & 3 \\ 4 & 1\end{array}\right)$ and $B=\left(\begin{array}{ll}x & 2 \\ y & 3\end{array}\right)$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 14

Solve for $x$ and $y$ such that $A B=B A$ if $A=\left(\begin{array}{rr}3 & x \\ -2 & 1\end{array}\right)$ and $B=\left(\begin{array}{cc}5 & 2 \\ y & 1\end{array}\right)$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 15

Solve for $x$ such that $A B=B A$ if $A=\left(\begin{array}{ccc}1 & 2 & 3 \\ x & 2 & -3 \\ 1 & 0 & 4\end{array}\right)$ and $B=\left(\begin{array}{ccc}-8 & x+3 & 12 \\ 23 & x-6 & -18 \\ 2 & -2 & 8\end{array}\right)$

SS

Sarvesh Somasundaram

Numerade Educator

Problem 16

Solve for $x$ and $y$ such that $A B=B A$ if $A=\left(\begin{array}{ccc}y & 2 & y+2 \\ x & 2 & -3 \\ 1 & y-1 & 4\end{array}\right)$ and $B=\left(\begin{array}{ccc}-8 & x+3 & 12 \\ 23 & x-6 & -18 \\ 2 & -2 & 8\end{array}\right)$

SS

Sarvesh Somasundaram

Numerade Educator