Secondary 4 Elementary Mathematics Vectors Matrices Quiz

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Secondary 4 Elementary Mathematics Quiz - Vectors Matrices

Name: ________________________
Class: ________________________
Date: ________________________
Score: ________ / 45

Duration: 60 Minutes
Total Marks: 45

Instructions:

• Answer all questions.
• Show all necessary working.
• Use a calculator where necessary.
• Give your answers in the spaces provided.

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Section A: Matrices (Questions 1–10)

1. Given matrix $A = \begin{pmatrix} 3 & -1 \\ 2 & 5 \end{pmatrix}$ and matrix $B = \begin{pmatrix} 0 & 4 \\ -3 & 1 \end{pmatrix}$, find $A + B$. [2]

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2. If $M = \begin{pmatrix} 4 & 7 \\ -2 & 0 \end{pmatrix}$, calculate $3M$. [2]

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3. Given $P = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ and $Q = \begin{pmatrix} 5 & 0 \\ -1 & 2 \end{pmatrix}$, find $P - Q$. [2]

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4. Find the product of $\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ and $\begin{pmatrix} 5 \\ 2 \end{pmatrix}$. [2]

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5. Given $R = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ and $S = \begin{pmatrix} 4 & 1 \\ 2 & 1 \end{pmatrix}$, calculate $RS$. [3]

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6. Given $R = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ and $S = \begin{pmatrix} 4 & 1 \\ 2 & 1 \end{pmatrix}$, calculate $SR$. [3]

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7. State whether $RS = SR$ for the matrices in questions 5 and 6. Justify your answer. [1]

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8. A matrix $T = \begin{pmatrix} x & 4 \\ 2 & 3 \end{pmatrix}$ is equal to $\begin{pmatrix} 7 & 4 \\ 2 & 3 \end{pmatrix}$. Find the value of $x$. [1]

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9. Given $A = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix}$, find a matrix $B$ such that $A + B = \begin{pmatrix} 5 & 2 \\ 4 & 8 \end{pmatrix}$. [2]

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10. A shop sells two types of fruit, Apples and Oranges. The quantities sold on Monday and Tuesday are represented by matrix $Q = \begin{pmatrix} 15 & 20 \\ 10 & 25 \end{pmatrix}$ (rows are days, columns are fruits). The prices per unit are $\begin{pmatrix} 1.20 \\ 0.80 \end{pmatrix}$. Find the total revenue for Monday. [3]

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Section B: Vectors (Questions 11–20)

11. Given $\mathbf{a} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}$, find the magnitude $|\mathbf{a}|$. [2]

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12. If $\vec{AB} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$ and $\vec{BC} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}$, find $\vec{AC}$. [2]

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13. Given $\vec{OA} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}$ and $\vec{OB} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}$, find $\vec{AB}$. [2]

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14. Find the vector $\vec{XY}$ if $X$ is $(2, 3)$ and $Y$ is $(5, -1)$. [2]

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15. Given $\mathbf{u} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$, find the vector $3\mathbf{u}$. [2]

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16. If $\vec{PQ} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}$, find the vector $\vec{QP}$. [1]

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17. Given $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}$, express $\mathbf{c} = 2\mathbf{a} + 3\mathbf{b}$ as a column vector. [3]

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18. Point $M$ is the midpoint of line segment $AB$. If $\vec{OA} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$ and $\vec{OB} = \begin{pmatrix} 8 \\ 10 \end{pmatrix}$, find $\vec{OM}$. [3]

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19. Given $\vec{AB} = \mathbf{a}$ and $\vec{BC} = 2\mathbf{a}$, find $\vec{AC}$ in terms of $\mathbf{a}$. [2]

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20. A vector $\mathbf{v} = \begin{pmatrix} k \\ 12 \end{pmatrix}$ has a magnitude of 13. Find the two possible values of $k$. [3]

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Secondary 4 Elementary Mathematics Quiz - Vectors Matrices (Answers)

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

2. $\begin{pmatrix} 3(4) & 3(7) \\ 3(-2) & 3(0) \end{pmatrix} = \begin{pmatrix} 12 & 21 \\ -6 & 0 \end{pmatrix}$ [2 marks]

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

4. $\begin{pmatrix} 2(5) + 3(2) \\ 1(5) + 4(2) \end{pmatrix} = \begin{pmatrix} 10+6 \\ 5+8 \end{pmatrix} = \begin{pmatrix} 16 \\ 13 \end{pmatrix}$ [2 marks]

5. $RS = \begin{pmatrix} 1(4)+2(2) & 1(1)+2(1) \\ 0(4)+3(2) & 0(1)+3(1) \end{pmatrix} = \begin{pmatrix} 8 & 3 \\ 6 & 3 \end{pmatrix}$ [3 marks]

6. $SR = \begin{pmatrix} 4(1)+1(0) & 4(2)+1(3) \\ 2(1)+1(0) & 2(2)+1(3) \end{pmatrix} = \begin{pmatrix} 4 & 11 \\ 2 & 7 \end{pmatrix}$ [3 marks]

7. No, $RS \neq SR$. Matrix multiplication is not commutative. [1 mark]

8. $x = 7$ [1 mark]

9. $B = \begin{pmatrix} 5-2 & 2-0 \\ 4-1 & 8-3 \end{pmatrix} = \begin{pmatrix} 3 & 2 \\ 3 & 5 \end{pmatrix}$ [2 marks]

10. Revenue = 15(1.20) + 20(0.80) = 18 + 16 = \34$ [3 marks]

11. $|\mathbf{a}| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ [2 marks]

12. $\vec{AC} = \vec{AB} + \vec{BC} = \begin{pmatrix} 2+4 \\ 5-1 \end{pmatrix} = \begin{pmatrix} 6 \\ 4 \end{pmatrix}$ [2 marks]

13. $\vec{AB} = \vec{OB} - \vec{OA} = \begin{pmatrix} -1-6 \\ 5-2 \end{pmatrix} = \begin{pmatrix} -7 \\ 3 \end{pmatrix}$ [2 marks]

14. $\vec{XY} = \begin{pmatrix} 5-2 \\ -1-3 \end{pmatrix} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}$ [2 marks]

15. $3\mathbf{u} = \begin{pmatrix} 3(2) \\ 3(3) \end{pmatrix} = \begin{pmatrix} 6 \\ 9 \end{pmatrix}$ [2 marks]

16. $\vec{QP} = -\vec{PQ} = \begin{pmatrix} -4 \\ -6 \end{pmatrix}$ [1 mark]

17. $\mathbf{c} = 2\begin{pmatrix} 1 \\ 2 \end{pmatrix} + 3\begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 2 \\ 4 \end{pmatrix} + \begin{pmatrix} 9 \\ -3 \end{pmatrix} = \begin{pmatrix} 11 \\ 1 \end{pmatrix}$ [3 marks]

18. $\vec{OM} = \frac{1}{2}(\vec{OA} + \vec{OB}) = \frac{1}{2}\begin{pmatrix} 2+8 \\ 4+10 \end{pmatrix} = \frac{1}{2}\begin{pmatrix} 10 \\ 14 \end{pmatrix} = \begin{pmatrix} 5 \\ 7 \end{pmatrix}$ [3 marks]

19. $\vec{AC} = \vec{AB} + \vec{BC} = \mathbf{a} + 2\mathbf{a} = 3\mathbf{a}$ [2 marks]

20. $\sqrt{k^2 + 12^2} = 13 \implies k^2 + 144 = 169 \implies k^2 = 25 \implies k = \pm 5$ [3 marks]