Sec 4 E Maths: Matrices, practice questions & worked solutions

Given that $\mathbf A=\begin{pmatrix}4&6\\0&-2\end{pmatrix}$ and $\mathbf B=\begin{pmatrix}2&k\\0&-1\end{pmatrix}$, find

1. $\mathbf A^2$, [1]

2. the value of $k$ if $\mathbf A=2\mathbf B$. [1]

The table below shows a multiple-choice test Sam and Roger took.

1. Represent this information in a $2\times3$ matrix, $\mathbf{S}$. [1]

2. The marks are awarded as follow: Correct $=2$ marks
   No attempt $=0$ mark
   Incorrect $=-1$ mark Represent the information in a $3\times1$ matrix, $\mathbf{T}$.

   Evaluate $\mathbf{ST}$ [2]

3. Explain what your answer to (b) represents. [1]

At a school carnival, two different classes sold ice cream to raise funds.
They sold in three different flavours: almond, blueberry and chocolate.
Class A sold 60 almond ice cream, 80 blueberry and 50 chocolate.
Class B sold 50 almond ice cream, 90 blueberry and 70 chocolate.

This information can be represented by the matrix $\mathbf P=\begin{pmatrix}60&80&50\\50&90&70\end{pmatrix}$

1. The almond ice cream is sold at $1.20.
   The blueberry ice cream is sold at $1.50.
   The chocolate ice cream is sold at $1.80.
   Represent these amounts in a $3\times1$ column matrix $\mathbf Q$. [1]

2. Evaluate the matrix $\mathbf T=\mathbf{PQ}$. [2]

3. State what each of the elements of $\mathbf T$ represents. [1]

4. Class A decides to donate 80% of their amount raise to an elderly home.
   Class B decides to donate 90% of their amount to the same elderly home.
   Write a matrix $\mathbf D$ such that the product $\mathbf{DT}$ will give the total amount donated. [1]

On a particular week, a 40-seater bus ferries 10 adults and 30 children to their destination every morning from Monday to Friday.
It ferries 20 adults and 10 children every afternoon from Monday to Friday.
The information may be represented by the matrix $\mathbf C=\begin{pmatrix}10&30\\20&10\end{pmatrix}$.

On Saturday morning, it ferries 25 adults and 15 children and in that afternoon, it ferries 10 adults and 30 children.

1. Represent the number of passengers carried on Saturday in a 2x2 matrix $\mathbf D$. [1]

2. Evaluate the matrix E = 5C + D. [2]

3. State what the elements of $\mathbf E$ represent. [1]

4. The fare for an adult is $2.50 and the fare for a child is $1.50.

   1. Evaluate the matrix F = C$\begin{pmatrix}250\\150\end{pmatrix}$. [1]

   2. State what the elements of $\mathbf F$ represent. [1]

A bakery produces cheesecakes, fruit cakes and red velvet cakes The number of cakes sold on a particular weekend is given by the matrix $G$.

$$\begin{array}{rcl}&\begin{array}{ccc}\text{cheese}&\text{fruit}&\text{red velvet}\end{array}&\\G=&\begin{pmatrix}15&20&14\\12&18&10\end{pmatrix}&\begin{array}{l}\text{Saturday}\\\text{Sunday}\end{array}\end{array}$$

A cheesecake is sold for $$x$, a fruit cake is sold for $$(x-8)$ and a red velvet cake is sold for $$(x-3)$.

1. Represent this information in a $3\times1$ matrix, $J$. [1]

2. Find in terms of $x$, the matrix $K=GJ$. [2]

3. State what the elements of $K$ represent. [1]

4. The total amount of money collected by the bakery from the sales of all the cakes on this particular weekend is $5676, find the value of $x$. [2]

In Shop $P$, a tub of ice cream cost $18.70, a carton of milk cost $8.60 and a box of chocolates cost $10.
In Shop $G$, a tub of ice cream cost $17.50, a carton of milk cost $7.40 and a box of chocolates cost $13.

The items are exactly the same in both shops.

The information above can be represented by the Matrix $\mathbf A=\begin{pmatrix}18.7&17.5\\8.6&7.4\\10&13\end{pmatrix}$.

1. Tina would like to buy 5 tubs of ice cream, 3 cartons of milk and 1 box of chocolates.
   Mike would like to buy 3 tubs of ice cream, 4 cartons of milk and $x$ boxes of chocolates.

   Represent this information in a $2\times3$ matrix $\mathbf B$. [1]

2. Find, in terms of $x$, the matrix $\mathbf C=\mathbf{BA}$. [2]

3. State what the elements in matrix $\mathbf C$ represents. [1]

4. Determine if it is possible for Mike to spend the same amount regardless of which shop he buys the items from. Justify your answer with calculations. [2]

A paint shop mixes three base colours (Red, Blue, Yellow) to create three shades, A, B, and C.
1 can of Shade A needs 2 litres of red, 3 litres of blue and 1 litre yellow base colours.
1 can of Shade B needs 1 litre of red, 2 litres of blue and 4 litres of yellow base colours.
1 can of Shade C needs 3 litres of red, 1 litre of blue and 2 litres of yellow base colours.

The matrix $M$ shows the number of litres of each base colour needed for 1 can of each shade.

$$M=\begin{array}{cl}\begin{array}{ccc}A&B&C\end{array}&\\[-3pt]\begin{pmatrix}2&1&3\\3&2&1\\1&4&2\end{pmatrix}&\begin{array}{l}\text{Red}\\\text{Blue}\\\text{Yellow}\end{array}\end{array}$$

1. The shop wants to make 4 cans of Shade A, 2 cans of Shade B, and $x$ cans of Shade C.
   Represent this information in a $3\times1$ column matrix, $N$, in terms of $x$. [1]

2. Find, in terms of $x$, the matrix $L=MN$. [2]

3. State what each of the element in matrix $L$ represents. [1]

4. The shop has 25 litres of red, 25 litres of blue, and 25 litres of yellow base colours in stock.
   Find the largest integer value of $x$ so that none of the base colours runs out. [2]

Starkids Centre is organising a talent competition for children. Children can choose to compete in 3 categories: singing, dancing and acting.

Matrix $\mathbf E$ shows the number of boys and girls in each category for the age group 5 to 8 years old.

$$\begin{array}{ccl}&\begin{array}{cc}\text{Boys}&\text{Girls}\end{array}&\\\mathbf E=&\begin{pmatrix}5&8\\7&4\\6&7\end{pmatrix}&\begin{array}{l}\text{Singing}\\\text{Dancing}\\\text{Acting}\end{array}\end{array}$$

Matrix $\mathbf F$ shows the number of boys and girls in each category for the age group 9 to 12 years old.

$$\begin{array}{ccl}&\begin{array}{cc}\text{Boys}&\text{Girls}\end{array}&\\\mathbf F=&\begin{pmatrix}12&9\\8&10\\15&11\end{pmatrix}&\begin{array}{l}\text{Singing}\\\text{Dancing}\\\text{Acting}\end{array}\end{array}$$

1. Evaluate the matrix $\mathbf T=\mathbf E+\mathbf F$. [1]

2. Each child is charged a registration fee for the competition. The registration is $30 for singing, $25 for dancing and $22 for acting.
   Represent the fees in a $1\times3$ matrix $\mathbf C$. [1]

3. Evaluate the matrix $\mathbf M=\mathbf{CT}$ and state what the elements in matrix $\mathbf M$ represent. [3]

4. Using matrix multiplication, calculate the total amount of registration fees collected for this competition. [2]

5. Starkids Centre decides to organise a similar competition for teenagers aged 13 to 18 years old. As compared to the competition for children aged 5 to 8 years old, the registration fees for teenagers is increased by 10% for singing, 20% for dancing and 15% for acting.
   Using matrix multiplication, calculate the registration fees for teenagers for each category.
   Give your answer in matrix form. [2]

1. On average, fruit seller $A$ is able to sell 20 mangoes, 32 apples and 15 oranges in a day.
   On average, fruit seller $B$ is able to sell 15 mangoes, 35 apples and 18 oranges in a day.

   The information can be represented by the matrix $\mathbf F=\begin{pmatrix}20&32&15\\15&35&18\end{pmatrix}$.

   1. Evaluate the matrix $\mathbf T=5\mathbf F\begin{pmatrix}1\\1\\1\end{pmatrix}$. [2]

   2. State what each of the elements of $\mathbf T$ represents. [1]

2. Seller $A$ sells a mango for 80 cents, an apple for 50 cents and an orange for 60 cents.
   Seller $B$ sells a mango for 70 cents, an apple for 40 cents and an orange for 50 cents.

   1. Represent this information in a $3\times2$ matrix $\mathbf C$. [1]

   2. Evaluate the matrix $\mathbf P=\mathbf{FC}$. [2]

   3. The cost price of all the fruits sold is $10 for each seller.

      Explain which seller made a higher profit. [1]

3. In a promotion, both seller $A$ and $B$ decide to give a discount of 20% for each mango and 10% for each apple.

   With the use of matrix multiplication, find a $3\times2$ matrix $\mathbf N$ where each element represents the new selling price of a mango, an apple and an orange by seller $A$ and $B$ respectively. [2]

Henry and John have time deposit in Australian dollars (AUD), New Zealand dollars (NZD) and American dollars (USD) with a bank. Their amounts deposited are shown in the matrix $\mathbf M$.

$$\mathbf M=\begin{array}{cl}\begin{array}{ccc}\text{AUD}&\text{NZD}&\text{USD}\end{array}&\\\begin{pmatrix}100000&150000&200000\\120000&180000&150000\end{pmatrix}&\begin{array}{l}\text{Henry}\\\text{John}\end{array}\end{array}$$

1. The exchange rates of these currency with Singapore dollars (SGD) are as follows.

   $$\begin{aligned}1\text{ AUD}&=0.83\text{ SGD}\\1\text{ NZD}&=0.77\text{ SGD}\\1\text{ USD}&=1.28\text{ SGD}\end{aligned}$$

   Write a $3\times1$ matrix $\mathbf E$ to represent these exchange rates. [1]

2. Evaluate the matrix $\mathbf D=\mathbf{ME}$. [2]

3. Explain what the elements in $\mathbf D$ represent. [1]

4. The annual interest rates for three deposits are as follows.

   $$\text{AUD: }1.18\%,\quad\text{NZD: }2.74\%,\quad\text{USD: }0.375\%$$

   1. Evaluate

      $$\mathbf A=\mathbf M\begin{pmatrix}1.18\%&0&0\\0&2.74\%&0\\0&0&0.375\%\end{pmatrix}$$

      [2]

   2. Explain what the elements in $\mathbf A$ represent. [1]

   3. Using matrix multiplication of the result in part d(i) and matrix $\mathbf E$, find the total annual interest in Singapore dollars that Henry and John each earned. [3]