Two subjects, Biology and Chemistry, are offered in the first year of a university science course. 

The matrix `N` lists the number of students enrolled in each subject.

`N = [(460),(360)]{:(text(Biology)),(text(Chemistry)):}`

The matrix `P` lists the proportion of these students expected to be awarded an `A`, `B`, `C`, `D` or `E` grade in each subject.

`{:((qquadqquadqquadA,qquad\ B,qquadquadC,qquad\ D,qquadE)),(P = [(0.05,0.125,0.175,0.45,0.20)]):}`

1. Write down the order of matrix `P`.  (1 mark)
2. Let the matrix  `R = NP`.

    

   1. Evaluate the matrix `R`.  (1 mark)
   2. Explain what the matrix element `R_24`  represents.  (1 mark)
3. Students enrolled in Biology have to pay a laboratory fee of $110, while students enrolled in Chemistry pay a laboratory fee of $95.

    

   1. Write down a clearly labelled row matrix, called `F`, that lists these fees.  (1 mark)
   2. Show a matrix calculation that will give the total laboratory fees, `L`, paid in dollars by the students enrolled in Biology and Chemistry. Find this amount.  (1 mark)

Rosa uses the following six-digit pin number for her bank account: 216342. 

With her knowledge of matrices, she decides to use matrix multiplication to disguise this pin number.

First she writes the six digits in the 2 × 3 matrix `A`.

`A = [(2,6,4),(1,3,2)]`

Next she creates a new matrix by forming the matrix product, `C = BA`,

where `B = [(1,-1),(2,-1)]`

1. 1. Determine the matrix `C = BA`.  (1 mark)
   2. From the matrix `C`, Rosa is able to write down a six-digit number that disguises her original pin number. She uses the same pattern that she used to create matrix `A` from the digits 216342.
      

       

      Write down the new six-digit number that Rosa uses to disguise her pin number.  (1 mark)

2. Show how the original matrix `A` can be regenerated from matrix `C`.  (1 mark) 

Matrix `F` below shows the flight connections for an airline that serves four cities, Anvil (`A`), Berga (`B`), Cantor (`C`) and Dantel (`D`).

`{:(qquad qquad qquad qquad quad text(from)),((qquad qquad quad\ A,B,C,D)),(F = [(0\ ,1\ ,0\ ,0),(1,0,1,0),(0,0,0,1),(0,1,0,0)]):}{:(),(),(A),(B),(C),(D):}{:(),(qquad text(to)):}`

In this matrix, the `1` in column `C` row `B`, for example, indicates that, using this airline, you can fly directly from Cantor to Berga. The `0` in column `C` row `D`, for example, indicates that you cannot fly directly from Cantor to Dantel.

1. Complete the following sentence. 
   

    

   On this airline, you can fly directly from Berga to ________ and ________.  (1 mark)

2. List the route that you must follow to fly from Anvil to Cantor.  (1 mark)
3. Evaluate the matrix product `G = KF`, where `K = [1,1,1,1]`.  (1 mark)

    

4. In the context of the problem, what information does matrix `G` contain?  (1 mark)

The ability level of students is assessed regularly and classified as beginner (`B`), intermediate (`I`) or advanced (`A`).

After each assessment, students either stay at their current level or progress to a higher level.

Students cannot be assessed at a level that is lower than their current level.

The expected number of students at each level after each assessment can be determined using the transition matrix, `T_1`, shown below.

`{:(qquad qquad text(before assessment)), (qquad qquad qquad quad {:(B, qquad quad I, quad A):}), (T_1 = [(0.50, 0, 0), (0.48, 0.80, 0), (0.02, 0.20, 1)] {:(B), (I), (A):} qquad text(after assessment)):}`

1. The element in the third row and third column of matrix `T_1` is the number 1.
    
   Explain what this tells you about the advanced-level students.  (1 mark)

Let matrix `S_n` be a state matrix that lists the number of students at beginner, intermediate and advanced levels after `n` assessments.

The number of students in the school, immediately before the first assessment of the year, is shown in matrix `S_0` below.

`S_0 = [(20), (60), (40)] {:(B), (I), (A):}`

2. 1. Write down the matrix `S_1` that contains the expected number of students at each level after one assessment.

       

       

      Write the elements of this matrix correct to the nearest whole number.  (1 mark)

   2. How many intermediate-level students have become advanced-level students after one assessment?  (1 mark)

Students in a music school are classified according to three ability levels: beginner (`B`), intermediate (`I`) or advanced (`A`).

Matrix `S_0`, shown below, lists the number of students at each level in the school for a particular week.

`S_0 = [(20), (60), (40)] {:(B), (I), (A):}`

1. How many students in total are in the music school that week?  (1 mark)

The music school has four teachers, David (`D`), Edith (`E`), Flavio (`F`) and Geoff (`G`).

Each teacher will teach a proportion of the students from each level, as shown in matrix `P` below.

`{:(qquadqquadqquad{:(Dquad,Eqquad,Fqquad,G):}),(P = [(0.25,0.5,0.15,0.1)]):}`

The matrix product, `Q = S_0P`, can be used to find the number of students from each level taught by each teacher.

2. 1. Complete matrix `Q`, shown below, by writing the missing elements in the shaded boxes.  (1 mark)

            

   2. How many intermediate students does Edith teach?  (1 mark)

The music school pays the teachers $15 per week for each beginner student, $25 per week for each intermediate student and $40 per week for each advanced student.

These amounts are shown in matrix `C` below.

`{:(qquad qquad quad{:(B quad, I quad,A):}),(C = [(15, 25, 40)]):}`

The amount paid to each teacher each week can be found using a matrix calculation.

3. 1. Write down a matrix calculation in terms of `Q` and `C` that results in a matrix that lists the amount paid to each teacher each week.  (1 mark)
   2. How much is paid to Geoff each week?  (1 mark)