Matrices and Systems of Equations — Topic Review

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This review covers all four lessons in the Matrices and Systems of Equations topic: Matrix Operations and Properties, Determinants and Inverses, Geometric Transformations with Matrices, and Solving Systems with Matrices. Questions are exam-style and increase in difficulty.

Mixed Review Questions

1. Fluency

   Q1 — Matrix Addition and Scalar Multiplication

   Let A = [[2, −1], [3, 4]] and B = [[0, 5], [−2, 1]]. Calculate: (a) A + B    (b) 3A    (c) 2A − B

2. Fluency

   Q2 — Matrix Multiplication

   Let P = [[1, 2], [0, 3]] and Q = [[4, −1], [2, 5]]. Calculate PQ and QP. Are they equal?

3. Fluency

   Q3 — Determinant of a 2×2 Matrix

   Find the determinant of each matrix: (a) A = [[5, 2], [3, 1]]    (b) B = [[−4, 6], [2, −3]]

4. Fluency

   Q4 — Inverse of a 2×2 Matrix

   Find the inverse of A = [[3, 1], [5, 2]].

5. Fluency

   Q5 — Reflection Matrix

   State the transformation matrix for a reflection in the x-axis, and apply it to the point (3, −2).

6. Understanding

   Q6 — Properties of Matrix Operations

   Given A = [[1, 0], [2, −1]], verify that (A²)^T = (A^T)². Is this always true?

7. Understanding

   Q7 — Determinant of a 3×3 Matrix

   Calculate det(C) where C = [[2, 1, 0], [3, −1, 2], [1, 4, −1]].

8. Understanding

   Q8 — Rotation Matrix

   A rotation of 90° anticlockwise about the origin is applied to the triangle with vertices A(1, 0), B(3, 0), C(3, 2). Find the images A′, B′, C′.

9. Understanding

   Q9 — Solving a System with Inverse Matrices

   Solve the system using the inverse matrix method:
   2x + 3y = 7
   x + 2y = 4

10. Understanding

    Q10 — Dilation Matrix

    Describe the geometric transformation given by T = [[3, 0], [0, 3]], and find the image of the line y = 2x − 1 under this transformation.

11. Problem Solving

    Q11 — Solving a 3×3 System by Row Reduction

    Solve the system using row reduction (Gaussian elimination):
    x + 2y − z = 3
    2x − y + z = 1
    3x + y + 2z = 7

12. Problem Solving

    Q12 — Composite Transformations

    Find the single matrix that represents: first a reflection in the y-axis, then a rotation of 90° anticlockwise about the origin. Apply this composite to the point (2, 1).

13. Problem Solving

    Q13 — Singular Systems and Geometric Interpretation

    Analyse the system 2x + 4y = 6 and x + 2y = k for values of k that give (a) no solution (b) infinitely many solutions. Explain geometrically.

14. Problem Solving

    Q14 — Finding a Matrix from Transformation Properties

    A linear transformation T maps (1, 0) to (3, 1) and (0, 1) to (−2, 4). Find the matrix of T and the image of the point (5, −3).

15. Problem Solving

    Q15 — Network and Matrix Application

    Three siblings share pocket money. Each week: Alex gives 20% of his money to Beth and 10% to Cal; Beth gives 30% of her money to Alex and 10% to Cal; Cal gives 20% of her money to Alex and 30% to Beth. Set up the transition matrix and find each sibling’s share after one week if they start with $100, $80, $60 respectively.