Matrices — Topic Review

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15 questions covering all Matrices sub-topics: notation, operations, multiplication, determinants, inverses and solving systems. Click Show Solution to reveal full working.

1. Fluency

   Matrix M =

   1	5	−3
   4	0	2

   . (a) State the order of M. (b) Identify element m₁₂. (c) Identify element m₂₃.

2. Fluency

   Calculate 2A − B where A =

   3	−1
   0	2

   and B =

   1	4
   −2	1

3. Fluency

   Find AB where A =

   2	1
   0	3

   and B =

   1	2
   4	−1

4. Fluency

   Find det(A) for A =

   5	2
   3	1

5. Fluency

   Find A⁻¹ for A =

   3	1
   5	2

6. Understanding

   Explain why the matrix

   2	4
   1	2

   has no inverse. What does this mean when using the matrix method to solve a system?

7. Understanding

   Find the value of k such that

   k	3
   2	k

   is singular.

8. Understanding

   Solve using the matrix inverse method:
   3x + y = 7
   x + y = 3

9. Understanding

   A 2×2 matrix P has det(P) = 4. What is det(2P)?

10. Understanding

    A store tracks stock of 2 products across 2 branches. Opening stock matrix S =

    30	20
    15	25

    . Sales matrix =

    12	8
    5	10

    . Find closing stock. Rows = Products A, B. Columns = Branches 1, 2.

11. Understanding

    Find AB and BA for A =

    1	0
    2	1

    and B =

    2	1
    0	3

    . Is AB = BA?

12. Problem Solving

    Solve using the matrix method, then verify by substitution:
    2x − y = 3
    x + 2y = 9

13. Problem Solving

    An event sells adult tickets ($12) and child tickets ($7). 80 tickets were sold for $760. Set up a matrix equation and solve to find how many of each ticket type were sold.

14. Problem Solving

    Given A =

    a	2
    1	3

    and det(A) = 4, find a. Then write down A⁻¹.

15. Problem Solving

    A coding matrix E =

    1	1
    0	1

    encodes message M =

    5

    3

    . (a) Find the encoded message EM. (b) The inverse of E is E⁻¹ =

    1	−1
    0	1

    . Decode the message

    9

    4

    .