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MATRICES, FUR1-NHT 2019 VCAA 4 MC

In a game between two teams, Hillside and Rovers, each team can score points in two ways.

The team may hit a Full or the team may hit a Bit.

More points are scored for hitting a Full than for hitting a Bit.

A team’s total point score is the sum of the points scored from hitting Fulls and Bits.

The table below shows the scores at the end of the game.
 


 

Let  `f`  be the number of points scored by hitting one Full.

Let  `b`  be the number of points scored by hitting one Bit.

Which one of the following matrix products can be evaluated to find the matrix  `[(f),(b)]`?

A. `\ [(4,8),(5,2)]^(−1) xx [(52),(49)]` B. `\ [(4,8),(5,2)] xx [(52),(49)]` C. `\ [(52,49)][(4,8),(5,2)]`
D. `\ [(52,49)][(4,8),(5,2)]^(−1)` E. `\ [(4,8,52),(5,2,49)]^(−1)`    
Show Answers Only

`A`

Show Worked Solution
`[(4,8),(5,2)][(f),(b)]` `= [(52),(49)]`
`[(f),(b)]` `= [(4,8),(5,2)]^(−1)[(52),(49)]`

 
`=>\ A`

MATRICES, FUR1 2018 VCAA 5 MC

Liam cycles, runs, swims and walks for exercise several times a month.

Each time he cycles, Liam covers a distance of `c` kilometres.

Each time he runs, Liam covers a distance of `r` kilometres.

Each time he swims, Liam covers a distance of `s` kilometres.

Each time he walks, Liam covers a distance of `w` kilometres.

The number of times that Liam cycled, ran, swam and walked each month over a four-month period, and the total distance that Liam travelled in each of those months, are shown in the table below.
 

 
The matrix that contains the distance each time Liam cycled, ran, swam and walked, `[(c),(r),(s),(w)]`, is

A. `[(5),(6),(7),(5)]` B. `[(8),(6),(1),(9)]` C. `[(8),(6),(7),(9)]`
           
D. `[(8),(8),(9),(8)]` E. `[(4290),(4931),(4623),(4291)]`    
Show Answers Only

`B`

Show Worked Solution

`text(Simultaneous equations matrix:)`

♦ Mean mark 42%.

`[(5,7,6,8),(8,6,9,7),(7,8,7,6),(8,8,5,5)][(c),(r),(s),(w)] = [(160),(172),(165),(162)]`
 

`[(c),(r),(s),(w)] = [(5,7,6,8),(8,6,9,7),(7,8,7,6),(8,8,5,5)]^(−1)[(160),(172),(165),(162)] = [(8),(6),(1),(9)]`
 

`=> B`

MATRICES, FUR2 2017 VCAA 1

A school canteen sells pies (`P`), rolls (`R`) and sandwiches (`S`).

The number of each item sold over three school weeks is shown in matrix `M`.
 

`{:(qquadqquadqquadquadPqquadRqquadS),(M = [(35,24,60),(28,32,43),(32,30,56)]{:(text(week 1)),(text(week 2)),(text(week 3)):}):}`
 

  1. In total, how many sandwiches were sold in these three weeks?  (1 mark)

The element in row `i` and column `j` of matrix `M` is `m_(ij)`.

  1. What does the element `m_12` indicate?  (1 mark)
  2. Consider the matrix equation

    `[(35,24,60),(28,32,43),(32,30,56)] xx [(a),(b),(c)] = [(491.55),(428.00),(487.60)]`

    where `a` = cost of one pie, `b` = cost of one roll and `c` = cost of one sandwich.

    1. What is the cost of one sandwich?  (1 mark)
    2. The matrix equation below shows that the total value of all rolls and sandwiches sold in these three weeks is $915.60

           `L xx [(491.55),(428.00),(487.60)] = [915.60]`

      Matrix `L` in this equation is of order `1 × 3`.
       
      Write down matrix `L`.  (1 mark)
Show Answers Only
  1. `159`
  2. `text(It represents the number of rolls sold in week 1.)`
    1. `$3.80`
    2. `text(Matrix)\ L = [0,1,1]`
Show Worked Solution
a.    `text(Total sandwiches)` `= 60 + 43 + 56`
    `= 159`

 

b.   `m_12 = 24`

`text(It represents the number of rolls sold in week 1.)`

 

c.i.    `[(a),(b),(c)]` `= [(35,24,60),(28,32,43),(32,30,56)]^(−1)[(491.55),(428.00),(487.60)]`
    `= [(4.65),(4.20),(3.80)]`

 

`:.\ text(C)text(ost of 1 sandwich = $3.80)`

 

c.ii.   `text(Matrix)\ L = [0,1,1]`

MATRICES, FUR2 2006 VCAA 3

Market researchers claim that the ideal number of bookshops (`x`), sports shoe shops (`y`) and music stores (`z`) for a shopping centre can be determined by solving the equations

`2x + y + z = 12`

`x-y+z=1`

`2y-z=6`

  1. Write the equations in matrix form using the following template.  (1 mark)

     

     
    `qquad[(qquadqquadqquad),(),()][(quad),(quad),(quad)] = [(quad),(quad),(quad)]`
     

     

  2. Do the equations have a unique solution? Provide an explanation to justify your response.  (1 mark)
  3. Write down an inverse matrix that can be used to solve these equations.  (1 mark)
  4. Solve the equations and hence write down the estimated ideal number of bookshops, sports shoe shops and music stores for a shopping centre.  (1 mark)
Show Answers Only
  1.  
    `[(2,1,1),(1,-1,1),(0,2,-1)][(x),(y),(z)] = [(12),(1),(6)]`
  2.  `text(Yes. See worked solutions.)`
  3.  
    `[(2,1,1),(1,-1,1),(0,2,-1)]^(-1) = [(-1,3,2),(1,-2,-1),(2,-4,-3)]`
  4. `text(3 bookshops, 4 sports shoe shops, 2 music stores.)`
Show Worked Solution
a.    `[(2,1,1),(1,-1,1),(0,2,-1)][(x),(y),(z)] = [(12),(1),(6)]`
♦ Mean mark 35% for all parts (combined).

 

b.    `text(det)\ [(2,1,1),(1,-1,1),(0,2,-1)] = 1 != 0`

 
`:.\ text(A unique solution exists.)`

 

c.   `text(By CAS,)`

`[(2,1,1),(1,-1,1),(0,2,-1)]^(-1) = [(-1,3,2),(1,-2,-1),(2,-4,-3)]`

 

d.    `[(x),(y),(z)]` `= [(-1,3,2),(1,-2,-1),(2,-4,-3)][(12),(1),(6)]`
    `= [(3),(4),(2)]`

 

`:.\ text(Estimated ideal numbers are:)`

`text(3 bookshops)`

`text(4 shoe shops)`

`text(2 music stores)`

MATRICES, FUR2 2007 VCAA 1

The table below displays the energy content and amounts of fat, carbohydrate and protein contained in a serve of four foods: bread, margarine, peanut butter and honey.
 

MATRICES, FUR2 2007 VCAA 1
 

  1. Write down a 2 x 3 matrix that displays the fat, carbohydrate and protein content (in columns) of bread and margarine.  (1 mark)
  2. `A` and `B` are two matrices defined as follows.
      
         `A = [(2,2,1,1)]`     `B = [(531),(41),(534),(212)]`
     

    1. Evaluate the matrix product  `AB`.  (1 mark)
    2. Determine the order of matrix product  `BA`.  (1 mark)

       

      Matrix `A` displays the number of servings of the four foods: bread, margarine, peanut butter and honey, needed to make a peanut butter and honey sandwich.

       

      Matrix `B` displays the energy content per serving of the four foods: bread, margarine, peanut butter and honey.

    3. Explain the information that the matrix product `AB` provides.  (1 mark)
  3. The number of serves of bread (`b`), margarine (`m`), peanut butter (`p`) and honey (`h`) that contain, in total, 53 grams of fat, 101.5 grams of carbohydrate, 28.5 grams of protein and 3568 kilojoules of energy can be determined by solving the matrix equation
      
         `[(1.2,6.7,10.7,0),(20.1,0.4,3.5,12.5),(4.2,0.6,4.6,0.1),(531,41,534,212)][(b),(m),(p),(h)] = [(53),(101.5),(28.5),(3568)]`
      
    Solve the matrix equation to find the values     `b`, `m`, `p` and `h`.  (2 marks)
Show Answers Only
  1.  
    `[(1.2,20.1,4.2),(6.7,0.4,0.6)]`
    1. `[1890]`
    2. `underset (4 xx 1) B xx underset (1 xx 4) A = underset (4 xx 4) (BA)`
    3. `BA\ text(provides the total energy)`
      `text(content of the servings of these)`
      `text(four foods in one sandwich.)`
  3. `b = 4, m = 4, p = 2, h = 1`
Show Worked Solution
a.    `[(1.2,20.1,4.2),(6.7,0.4,0.6)]`

 

b.i.    `AB` `= [(2, 2, 1, 1)] [(531), (41), (534), (212)]`
    `= [1890]`

 

b.ii.   `underset (4 xx 1) B xx underset (1 xx 4) A = underset (4 xx 4) (BA)`

 

b.iii.   `BA\ text(provides the total energy content of the)`
 

`text(servings of these four foods in one sandwich.)`

 

c.    `[(b),(m),(p),(h)]` `= [(1.2,6.7,10.7,0),(20.1,0.4,3.5,12.5),(4.2,0.6,4.6,0.1),(531,41,534,212)]^(-1)[(53),(101.5),(28.5),(3568)]`
    `= [(4),(4),(2),(1)]\ \ \ text{(by graphics calculator)}`

 

`:. b = 4, m = 4, p = 2\ text(and)\ h = 1.`

MATRICES, FUR2 2009 VCAA 2

Tickets for the function are sold at the school office, the function hall and online.

Different prices are charged for students, teachers and parents.

Table 1 shows the number of tickets sold at each place and the total value of sales.
  

MATRICES, FUR2 2009 VCAA 21
 

For this function

    • student tickets cost  `$x`
    • teacher tickets cost  `$y`
    • parent tickets cost  `$z`.
  1. Use the information in Table 1 to complete the following matrix equation by inserting the missing values in the shaded boxes.  (1 mark)
     
         MATRICES, FUR2 2009 VCAA 22
     
  2. Use the matrix equation to find the cost of a teacher ticket to the school function.  (2 marks)
Show Answers Only
  1. `text(35 and 2)`
  2. `$32`
Show Worked Solution

a.   `text(35 and 2)`

 

MARKER’S COMMENT: Simply writing the column matrix in part (b) did not earn full marks. Students must extract the required data.
b.    `[(x),(y),(z)]` `= [(283,28,5),(35,4,2),(84,3,7)]^(-1)[(8712),(1143),(2609)]`
    `= [(27),(32),(35)]`

 

`:.\ text(C)text(ost of a teacher ticket = $32)`

MATRICES, FUR2 2010 VCAA 3

The basketball coach has written three linear equations which can be used to predict the number of points, `p`, rebounds, `r`, and assists, `a`, that Oscar will have in his next game.

The equations are    `p + r + a` `= 33`
`2p - r + 3a` `= 40`
`p + 2r + a` `= 43`
  1. These equations can be written equivalently in matrix form.

     

    Complete the missing information below.  (1 mark)

 
`[(qquadqquad),(qquadqquad),(qquadqquad)][(p),(r),(a)] = [(33),(40),(43)]`
 

This matrix equation can be solved in the following way.

 
`[(p),(r),(a)] = [(7,-1,-4),(-1,0,1),(x,1,3)][(33),(40),(43)]`
 

  1. Determine the value of `x` shown in the matrix equation above.  (1 mark)
  2. How many rebounds is Oscar predicted to have in his next game?  (1 mark)
Show Answers Only
  1.  
    `[(1,1,1),(2,-1,3),(1,2,1)][(p),(r),(a)] = [(33),(40),(43)]`
  2. `-5`
  3. `10`
Show Worked Solution
a.    `[(1,1,1),(2,-1,3),(1,2,1)][(p),(r),(a)] = [(33),(40),(43)]`

 

b.    `[(7,-1,-4),(-1,0,1),(x,1,3)]` `=[(1,1,1),(2,-1,3),(1,2,1)]^(-1)`
    `= [(7,-1,-4),(-1,0,1),(-5,1,3)]`

 

`:.x = -5`

 

c.    `[(p),(r),(a)]` `= [(7,-1,-4),(-1,0,1),(-5,1,3)][(33),(40),(43)]`
    `= [(19),(10),(4)]`

 

 

`:.\ text(Oscar is predicted to have 10 rebounds in the next game.)`

MATRICES, FUR1 2008 VCAA 6 MC

The solution of the matrix equation  `[(0, – 3, 2), (1, 1, 1), (– 2, 0, 3)] [(x), (y), (z)] = [(11), (5), (8)]`  is

A.  `[(1), (24), (2)]` B.  `[(2), (– 1), (4)]`
       
C.  `[(2), (1), (3)]` D.  `[(– 11), (4/3), (8)]`
       
E.  `[(11), (5), (8)]`    

 

Show Answers Only

`B`

Show Worked Solution

`[(0, – 3, 2), (1, 1, 1), (– 2, 0, 3)] [(x), (y), (z)]= [(11), (5), (8)]`

`:. [(x), (y), (z)]` `= [(0, – 3, 2), (1, 1, 1), (– 2, 0, 3)]^-1 [(11), (5), (8)]`
  `= [(2), (– 1), (4)]`

 
`=>   B`

MATRICES, FUR1 2010 VCAA 3 MC

The total cost of one ice cream and three soft drinks at Catherine’s shop is $9.

The total cost of two ice creams and five soft drinks is $16.

Let `x` be the cost of an ice cream and `y` be the cost of a soft drink

The matrix `[(x), (y)]` is equal to
 

A.  `[(1, 3), (2, 5)] [(x), (y)]`

B.  `[(1, 3), (2, 5)] [(9), (16)]`

C.  `[(1, 2), (3, 5)] [(9), (16)]`

D.  `[(– 5, 2), (3, – 1)] [(9), (16)]`

E.  `[(– 5, 3), (2, – 1)] [(9), (16)]`

Show Answers Only

`E`

Show Worked Solution

`[(1, 3), (2, 5)][(x), (y)] = [(9), (16)]`

♦♦ Mean mark 35%.
`:. [(x), (y)]` `= [(1, 3), (2, 5)]^-1 [(9), (16)]`
  `= [(– 5, 3), (2, – 1)] [(9), (16)]`

 
`=>   E`

MATRICES, FUR1 2012 VCAA 3 MC

`x + z` `= 6`
`2y + z` `= 8`
`2x + y + 2z` `= 15`

 
The solution of the simultaneous equations above is given by

MATRICES, FUR1 2012 VCAA 3 MC ab1

MATRICES, FUR1 2012 VCAA 3 MC cd

MATRICES, FUR1 2012 VCAA 3 MC e

Show Answers Only

`A`

Show Worked Solution

`[(1,0,1),(0,2,1),(2,1,2)][(x),(y),(z)] = [(6),(8),(15)]`

♦ Mean mark 41%.
`:. [(x),(y),(z)]` `= [(1,0,1),(0,2,1),(2,1,2)]^(−1)[(6),(8),(15)]`
  `= [(−3,−1,2),(−2,0,1),(4,1,−2)][(6),(8),(15)]`

 
`rArr A`

MATRICES, FUR1 2013 VCAA 6 MC

A worker can assemble 10 bookcases and four desks in 360 minutes, and eight bookcases and three desks in 280 minutes.

If each bookcase takes `b` minutes to assemble and each desk takes `d` minutes to assemble, the matrix `[(b), (d)]` will be given by

A.   `[(-1.5, 2), (4, -5)][(360), (280)]`

B.   `[(10, 4), (8, 3)][(360), (280)]`

C.   `[(3, -4), (-8, 10)][(360), (280)]`

D.   `[(5,- 2), (-4, 1.5)][(360), (280)]`

E.   `[(10), (4)][360] + [(8), (3)][280]`

Show Answers Only

`A`

Show Worked Solution

`[(10,4),(8,3)][(b),(d)] = [(360),(280)]`

♦ Mean mark 41%.
`:. [(b),(d)]` `= [(10,4),(8,3)]^(−1)[(360),(280)]`
  `= 1/(10 xx 3 – 4 xx 8)[(3,−4),(−8,10)][(360),(280)]`
  `= −1/2[(3,−4),(−8,10)][(360),(280)]`
  `= [(−1.5,2),(4,−5)][(360),(280)]`

 
`rArr A`

MATRICES, FUR1 2015 VCAA 6 MC

A carpenter can make four coffee tables and seven stools in a total of 33 hours.

The carpenter can make two coffee tables and three pencil boxes in a total of 12 hours.

The carpenter can make five stools and one pencil box in a total of 10 hours.

The time, in hours, that it takes to make one coffee table is closest to

A.   2

B.   3

C.   4

D.   5

E.   6

Show Answers Only

`D`

Show Worked Solution

`[(4,7,0),(2,0,3),(0,5,1)][(C),(S),(P)] = [(33),(12),(10)]`

`[(C),(S),(P)]` `= [(4,7,0),(2,0,3),(0,5,1)]^(−1)[(33),(12),(10)]`
  `= [(4.99),(1.86),(0.68)]`

 

`:.\ text(One coffee table takes 5 hours)`

`=> D`