Secondary 4 Elementary Mathematics Vectors Matrices Quiz

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Secondary 4 Elementary Mathematics Quiz - Vectors Matrices

Name: ___________________________
Class: ____________
Date: ____________
Score: ____________

Duration: 60 minutes
Total Marks: 40

Instructions:

• Answer all 20 questions.
• Show your working clearly where required.
• Use the spaces provided to write your answers.
• Calculators may be used.

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Section A: Matrices – Display, Interpretation, and Scalar Multiplication (Questions 1–7)

1. [2 marks]
The table below shows the number of books sold by a shop on Monday and Tuesday.

	Fiction	Non-Fiction
Monday	12	8
Tuesday	15	10

Write this information as a $2 \times 2$ matrix $M$, with rows for Monday and Tuesday.

Answer:
$M = \begin{pmatrix} \quad & \quad \\ \quad & \quad \end{pmatrix}$

2. [1 mark]
Given the matrix $A = \begin{pmatrix} 3 & 5 \\ 2 & 7 \end{pmatrix}$, what does the element in the second row and first column represent if $A$ shows test scores of 2 students in English and Maths?

Answer: ________________________________________

3. [2 marks]
A matrix $B = \begin{pmatrix} 4 & 6 \\ 1 & 3 \end{pmatrix}$ shows the cost ($) of two items at two stalls. Find$2B$.

Answer:
$2B = \begin{pmatrix} \quad & \quad \\ \quad & \quad \end{pmatrix}$

4. [2 marks]
The matrix $C = \begin{pmatrix} -2 & 0 \\ 5 & -3 \end{pmatrix}$. Calculate $-3C$.

Answer:
$-3C = \begin{pmatrix} \quad & \quad \\ \quad & \quad \end{pmatrix}$

5. [1 mark]
A class has a matrix $D = \begin{pmatrix} 9 & 4 & 7 \end{pmatrix}$ showing the number of red, blue, and green files. How many columns does $D$ have?

Answer: ____________

6. [2 marks]
The weekly savings (in $) of three students are shown by$S = \begin{pmatrix} 10 & 15 & 20 \end{pmatrix}$. If each student saves triple the amount next week, write the new matrix.

Answer: ________________________________________

7. [2 marks]
Matrix $P = \begin{pmatrix} 2 & 1 \\ 0 & 4 \end{pmatrix}$ shows points earned by two players in two rounds. State the value of the element $p_{12}$ and explain what it means.

Answer: ________________________________________

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Section B: Matrix Addition, Subtraction, and Multiplication (Questions 8–14)

8. [2 marks]
$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, $B = \begin{pmatrix} 5 & 0 \\ 1 & 2 \end{pmatrix}$. Find $A + B$.

Answer:
$A + B = \begin{pmatrix} \quad & \quad \\ \quad & \quad \end{pmatrix}$

9. [2 marks]
Using the same $A$ and $B$ above, find $B - A$.

Answer:
$B - A = \begin{pmatrix} \quad & \quad \\ \quad & \quad \end{pmatrix}$

10. [3 marks]
$X = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$, $Y = \begin{pmatrix} 4 & 2 \\ 1 & 5 \end{pmatrix}$. Calculate $XY$.

Answer:
$XY = \begin{pmatrix} \quad & \quad \\ \quad & \quad \end{pmatrix}$

11. [2 marks]
$A = \begin{pmatrix} 3 & 2 \end{pmatrix}$, $B = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$. Find $AB$.

Answer:
$AB = \begin{pmatrix} \quad \end{pmatrix}$

12. [2 marks]
With the same $A$ and $B$ from Q11, explain why $BA$ cannot be found, or if it can, state its order.

Answer: ________________________________________

13. [3 marks]
$M = \begin{pmatrix} 1 & 0 & 2 \\ 3 & 1 & 1 \end{pmatrix}$, $N = \begin{pmatrix} 2 & 1 \\ 0 & 3 \\ 1 & 2 \end{pmatrix}$. Find $MN$.

Answer:
$MN = \begin{pmatrix} \quad & \quad \\ \quad & \quad \end{pmatrix}$

14. [2 marks]
Two classes recorded the number of students who brought umbrellas on two days: Class 4A: $\begin{pmatrix} 10 & 12 \end{pmatrix}$, Class 4B: $\begin{pmatrix} 8 & 9 \end{pmatrix}$. Find the total as a single matrix.

Answer: ________________________________________

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Section C: Vectors – Basics, Operations, and Geometry (Questions 15–20)

15. [1 mark]
Write the vector $\vec{AB}$ as a column vector if $A$ is at $(1,2)$ and $B$ is at $(4,6)$.

Answer: $\begin{pmatrix} \quad \\ \quad \end{pmatrix}$

16. [2 marks]
Given $\vec{u} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$ and $\vec{v} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$, find $\vec{u} + \vec{v}$.

Answer: $\begin{pmatrix} \quad \\ \quad \end{pmatrix}$

17. [2 marks]
Given $\vec{p} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}$, find $3\vec{p}$.

Answer: $\begin{pmatrix} \quad \\ \quad \end{pmatrix}$

18. [2 marks]
$\vec{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$, $\vec{b} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}$. Find $2\vec{a} - \vec{b}$.

Answer: $\begin{pmatrix} \quad \\ \quad \end{pmatrix}$

19. [3 marks]
In the diagram below, $O$ is the origin, $\vec{OA} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ and $\vec{OB} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}$. Find $\vec{AB}$ and the magnitude $|\vec{AB}|$.

Image pending generation: diagram for Q19.

Answer:
$\vec{AB} = \begin{pmatrix} \quad \\ \quad \end{pmatrix}$, $|\vec{AB}| =$ ____________

20. [3 marks]
$\vec{m} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$, $\vec{n} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$. Given that $\vec{r} = 2\vec{m} + \vec{n}$, find $\vec{r}$ and the unit vector in the direction of $\vec{r}$.

Answer:
$\vec{r} = \begin{pmatrix} \quad \\ \quad \end{pmatrix}$, unit vector = $\begin{pmatrix} \quad \\ \quad \end{pmatrix}$

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Answer Key: Secondary 4 Elementary Mathematics Quiz - Vectors Matrices

Total Marks: 40
Topic: Vectors & Matrices

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Section A: Matrices – Display, Interpretation, and Scalar Multiplication

Q1. [2 marks]
$M = \begin{pmatrix} 12 & 8 \\ 15 & 10 \end{pmatrix}$
Teaching note: A matrix is a rectangular array. Rows = Monday, Tuesday; columns = Fiction, Non-Fiction. 1 mark for correct rows, 1 mark for correct columns.

Q2. [1 mark]
The element is 2, representing Student 2’s English score.
Teaching note: Second row = second student, first column = English. Direct interpretation.

Q3. [2 marks]
$2B = 2 \times \begin{pmatrix} 4 & 6 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} 8 & 12 \\ 2 & 6 \end{pmatrix}$
Marking: 1 mark for multiplying each element, 1 mark for final matrix.

Q4. [2 marks]
$-3C = -3 \times \begin{pmatrix} -2 & 0 \\ 5 & -3 \end{pmatrix} = \begin{pmatrix} 6 & 0 \\ -15 & 9 \end{pmatrix}$
Common mistake: Forgetting negative sign on -3 × 5 = -15, then × -3 = +9.

Q5. [1 mark]
3 columns.
Teaching note: $D$ is a $1 \times 3$ matrix (1 row, 3 columns).

Q6. [2 marks]
New matrix = $3 \times \begin{pmatrix} 10 & 15 & 20 \end{pmatrix} = \begin{pmatrix} 30 & 45 & 60 \end{pmatrix}$
Teaching note: Scalar multiplication multiplies every entry.

Q7. [2 marks]
$p_{12} = 1$. It means Player 1 earned 1 point in Round 2.
Teaching note: Subscript row,column: $p_{12}$ = row 1, col 2.

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Section B: Matrix Addition, Subtraction, and Multiplication

Q8. [2 marks]
$A+B = \begin{pmatrix} 1+5 & 2+0 \\ 3+1 & 4+2 \end{pmatrix} = \begin{pmatrix} 6 & 2 \\ 4 & 6 \end{pmatrix}$

Q9. [2 marks]
$B-A = \begin{pmatrix} 5-1 & 0-2 \\ 1-3 & 2-4 \end{pmatrix} = \begin{pmatrix} 4 & -2 \\ -2 & -2 \end{pmatrix}$

Q10. [3 marks]
$XY = \begin{pmatrix} 2&1 \\ 0&3 \end{pmatrix} \begin{pmatrix} 4&2 \\ 1&5 \end{pmatrix}$
Row1: $(2\times4 + 1\times1,\; 2\times2 + 1\times5) = (9, 9)$
Row2: $(0\times4 + 3\times1,\; 0\times2 + 3\times5) = (3, 15)$
$XY = \begin{pmatrix} 9 & 9 \\ 3 & 15 \end{pmatrix}$
Marking: 1 mark each row correct.

Q11. [2 marks]
$AB = \begin{pmatrix} 3&2 \end{pmatrix} \begin{pmatrix} 1 \\ 4 \end{pmatrix} = (3\times1 + 2\times4) = (11)$
Result is $1\times1$ matrix: $\begin{pmatrix} 11 \end{pmatrix}$

Q12. [2 marks]
$BA$: $B$ is $2\times1$, $A$ is $1\times2$. Inner dimensions match (1=1), so $BA$ can be found; order is $2\times2$.
$BA = \begin{pmatrix} 1\\4 \end{pmatrix} \begin{pmatrix} 3&2 \end{pmatrix} = \begin{pmatrix} 3&2 \\ 12&8 \end{pmatrix}$
Teaching note: Matrix multiplication needs columns of first = rows of second.

Q13. [3 marks]
$M$ is $2\times3$, $N$ is $3\times2$ → $MN$ is $2\times2$.
Row1: $(1\times2+0\times0+2\times1,\; 1\times1+0\times3+2\times2) = (4, 5)$
Row2: $(3\times2+1\times0+1\times1,\; 3\times1+1\times3+1\times2) = (7, 8)$
$MN = \begin{pmatrix} 4 & 5 \\ 7 & 8 \end{pmatrix}$

Q14. [2 marks]
Total = $\begin{pmatrix} 10+8 & 12+9 \end{pmatrix} = \begin{pmatrix} 18 & 21 \end{pmatrix}$

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Section C: Vectors – Basics, Operations, and Geometry

Q15. [1 mark]
$\vec{AB} = \begin{pmatrix} 4-1 \\ 6-2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$

Q16. [2 marks]
$\vec{u}+\vec{v} = \begin{pmatrix} 3+1 \\ -2+4 \end{pmatrix} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$

Q17. [2 marks]
$3\vec{p} = 3 \begin{pmatrix} 5 \\ 1 \end{pmatrix} = \begin{pmatrix} 15 \\ 3 \end{pmatrix}$

Q18. [2 marks]
$2\vec{a} - \vec{b} = \begin{pmatrix} 4 \\ 6 \end{pmatrix} - \begin{pmatrix} -1 \\ 2 \end{pmatrix} = \begin{pmatrix} 5 \\ 4 \end{pmatrix}$

Q19. [3 marks]
$\vec{AB} = \vec{OB} - \vec{OA} = \begin{pmatrix} 4-2 \\ 3-1 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \end{pmatrix}$
$|\vec{AB}| = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}$
Marking: 1 mark vector, 2 marks magnitude with working.

Q20. [3 marks]
$\vec{r} = 2\begin{pmatrix} 3 \\ 2 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 6 \\ 4 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 5 \\ 8 \end{pmatrix}$
$|\vec{r}| = \sqrt{5^2+8^2} = \sqrt{89}$
Unit vector = $\frac{1}{\sqrt{89}}\begin{pmatrix} 5 \\ 8 \end{pmatrix} = \begin{pmatrix} 5/\sqrt{89} \\ 8/\sqrt{89} \end{pmatrix}$
Marking: 1 mark r, 2 marks unit vector.