Secondary 4 Elementary Mathematics Vectors Matrices Quiz

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Secondary 4 Elementary Mathematics Quiz - Vectors Matrices

Name: ___________________________
Class: ______________
Date: ______________
Score: _______ / 40

Duration: 50 minutes
Total Marks: 40

Instructions:

• Answer all 20 questions.
• Show your working clearly where required.
• Section A: 10 short questions (1 mark each). Section B: 6 written questions (2 marks each). Section C: 4 extended questions (3 marks each).
• Calculators may be used.

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Section A (Questions 1–10, 1 mark each)

1. Write the order of the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$.

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2. Given $A = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix}$, find $2A$.

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3. Add the matrices $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$.

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4. Subtract: $\begin{pmatrix} 9 & 8 \\ 7 & 6 \end{pmatrix} - \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$.

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5. A vector $\vec{v}$ has components $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$. Write the column vector for $-\vec{v}$.

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6. Given $\vec{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 4 \\ 0 \end{pmatrix}$, find $\vec{a} + \vec{b}$.

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7. Let $M = \begin{pmatrix} 2 & 1 \end{pmatrix}$ and $N = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$. State the order of the product $MN$.

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8. Multiply: $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 5 \\ 7 \end{pmatrix}$.

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9. A matrix shows the number of books read by 2 students in 3 subjects: $\begin{pmatrix} 4 & 2 & 1 \\ 3 & 0 & 5 \end{pmatrix}$. What does the number 5 represent?

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10. Given $\vec{p} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}$, find $3\vec{p}$.

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Section B (Questions 11–16, 2 marks each)

11. Given $X = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $Y = \begin{pmatrix} 0 & 1 \\ 2 & 3 \end{pmatrix}$, find $X + Y$.

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12. Find the product $AB$ where $A = \begin{pmatrix} 1 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 4 \\ 5 & 6 \end{pmatrix}$.

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13. A translation is represented by vector $\vec{t} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$. A point $P$ has position vector $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$. Find the position vector of the image of $P$ after translation by $\vec{t}$.

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14. The matrix below shows the cost ($) of 2 items at 3 shops:$\begin{pmatrix} 2 & 3 & 4 \ 5 & 6 & 7 \end{pmatrix}$ (row 1: pens, row 2: books). A student buys 2 pens and 1 book from shop 2. Write a matrix for the quantity bought and find the total cost using matrix multiplication.

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15. Given $\vec{u} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ and $\vec{v} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}$, find $2\vec{u} - \vec{v}$.

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16. Multiply the scalar $-3$ by the matrix $\begin{pmatrix} 1 & -2 \\ 0 & 4 \end{pmatrix}$.

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Section C (Questions 17–20, 3 marks each)

17. A factory makes 2 products (A and B) using 2 materials (X and Y). The matrix $U = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$ shows units of material per product (row = material, column = product). The matrix $C = \begin{pmatrix} 5 & 3 \end{pmatrix}$ shows cost per unit of each material. Find the cost matrix for producing one unit of each product and state the cost of product B.

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18. Given $\vec{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$, $\vec{b} = \begin{pmatrix} 3 \\ 0 \end{pmatrix}$, and $\vec{c} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$, find $2\vec{a} + \vec{b} - \vec{c}$ and describe the vector operation steps.

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19. Let $P = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $Q = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix}$. Find $PQ$ and state whether $QP$ would be equal to $PQ$.

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20. A survey of 3 classes (C1, C2, C3) on favourite sport (Football, Swimming) gives: $\begin{pmatrix} 12 & 8 \\ 10 & 9 \\ 7 & 11 \end{pmatrix}$. Interpret the entry in row 2 column 1, and find the total number of students who chose Swimming.

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Secondary 4 Elementary Mathematics Quiz - Vectors Matrices (Answer Key)

Total Marks: 40
Topic: Vectors & Matrices (Syllabus N9, G7 inferred)

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Section A Answers (1 mark each)

Q1. Order: $2 \times 3$ (2 rows, 3 columns).
Teaching note: Matrix order is written as rows × columns. Count rows top to bottom = 2, columns left to right = 3.

Q2. $2A = \begin{pmatrix} 4 & 0 \\ 2 & 6 \end{pmatrix}$
Method: Multiply each element by 2: $2\times2=4$, $2\times0=0$, $2\times1=2$, $2\times3=6$.

Q3. $\begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}$
Method: Add corresponding elements: $1+5=6$, $2+6=8$, $3+7=10$, $4+8=12$.

Q4. $\begin{pmatrix} 8 & 6 \\ 4 & 2 \end{pmatrix}$
Method: Subtract corresponding: $9-1=8$, $8-2=6$, $7-3=4$, $6-4=2$.

Q5. $-\vec{v} = \begin{pmatrix} -3 \\ 2 \end{pmatrix}$
Method: Reverse signs: $-(3)=-3$, $-(-2)=2$.

Q6. $\vec{a}+\vec{b} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}$
Method: Add components: $1+4=5$, $2+0=2$.

Q7. Order of $MN$ is $1 \times 1$
Method: $M$ is $1\times2$, $N$ is $2\times1$; product is $1\times1$ (outer dimensions).

Q8. $\begin{pmatrix} 5 \\ 7 \end{pmatrix}$
Method: Identity matrix leaves vector unchanged: $\begin{pmatrix}1&0\\0&1\end{pmatrix}\begin{pmatrix}5\\7\end{pmatrix}=\begin{pmatrix}5\\7\end{pmatrix}$.

Q9. Student 2 read 5 books in subject 3.
Teaching note: Row 2 = student 2, column 3 = subject 3.

Q10. $3\vec{p} = \begin{pmatrix} -3 \\ 9 \end{pmatrix}$
Method: Scalar multiply: $3\times(-1)=-3$, $3\times3=9$.

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Section B Answers (2 marks each)

Q11. $X+Y = \begin{pmatrix} 1 & 3 \\ 5 & 7 \end{pmatrix}$
Working: $\begin{pmatrix}1+0 & 2+1\\3+2 & 4+3\end{pmatrix}$ (1 mark for setup, 1 mark final).

Q12. $AB = \begin{pmatrix} 1\times3+2\times5 & 1\times4+2\times6 \end{pmatrix} = \begin{pmatrix} 13 & 16 \end{pmatrix}$
Working: Row × column: $3+10=13$, $4+12=16$ (1m method, 1m answer).

Q13. Image position = $\begin{pmatrix} 5 \\ 3 \end{pmatrix}$
Working: $\begin{pmatrix}3\\4\end{pmatrix}+\begin{pmatrix}2\\-1\end{pmatrix}=\begin{pmatrix}5\\3\end{pmatrix}$ (1m vector add, 1m result).

Q14. Quantity matrix $Q = \begin{pmatrix} 2 & 1 \end{pmatrix}$ (pens=2, books=1 from shop 2 column = $\begin{pmatrix}3\\6\end{pmatrix}$? Wait: shop 2 is column 2: pens=3, books=6; but student buys 2 pens,1 book → quantity row matrix $\begin{pmatrix}2 & 1\end{pmatrix}$ times cost column $\begin{pmatrix}3\\6\end{pmatrix}$ = $2\times3+1\times6=12$).
Answer: Total cost = \12$. *Working:* Cost column for shop2 =$\begin{pmatrix}3\6\end{pmatrix}$;$Q \times \text{cost} = 2(3)+1(6)=12$ (1m quantity, 1m total).

Q15. $2\vec{u}-\vec{v} = \begin{pmatrix} 4 \\ 2 \end{pmatrix} - \begin{pmatrix} -1 \\ 3 \end{pmatrix} = \begin{pmatrix} 5 \\ -1 \end{pmatrix}$
Working: $2\vec{u}=\begin{pmatrix}4\\2\end{pmatrix}$; subtract components (1m, 1m).

Q16. $-3 \begin{pmatrix} 1 & -2 \\ 0 & 4 \end{pmatrix} = \begin{pmatrix} -3 & 6 \\ 0 & -12 \end{pmatrix}$
Working: Each element × -3 (1m, 1m).

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Section C Answers (3 marks each)

Q17. Cost matrix = $C \times U = \begin{pmatrix}5&3\end{pmatrix}\begin{pmatrix}3&2\\1&4\end{pmatrix} = \begin{pmatrix} 5\times3+3\times1 & 5\times2+3\times4 \end{pmatrix} = \begin{pmatrix} 18 & 22 \end{pmatrix}$.
Product B cost = \22$.
Marks: 1m for setting multiplication, 1m for calculation, 1m for stating B.
Note: Row×row matrix times 2×2 gives 1×2 (cost per product).

Q18. $2\vec{a}+\vec{b}-\vec{c} = \begin{pmatrix}2\\4\end{pmatrix}+\begin{pmatrix}3\\0\end{pmatrix}-\begin{pmatrix}-1\\4\end{pmatrix} = \begin{pmatrix}5\\-4\end{pmatrix}-\begin{pmatrix}-1\\4\end{pmatrix} = \begin{pmatrix}6\\-8\end{pmatrix}$.
Steps: Scalar multiply, then add, then subtract component-wise.
Marks: 1m each for correct intermediate and final.

Q19. $PQ = \begin{pmatrix}1&2\\3&4\end{pmatrix}\begin{pmatrix}2&0\\1&2\end{pmatrix} = \begin{pmatrix}1\times2+2\times1 & 1\times0+2\times2\\3\times2+4\times1 & 3\times0+4\times2\end{pmatrix} = \begin{pmatrix}4&4\\10&8\end{pmatrix}$.
$QP \neq PQ$ generally (matrix multiplication not commutative).
Marks: 2m for PQ, 1m for stating non-equal.

Q20. Row2Col1 = Class C2 football favourites = 10 students. Total swimming = $8+9+11 = 28$.
Marks: 1m interpretation, 2m total (or 1m each column sum).
Teaching: Swimming is column 2; sum entries.