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IB Mathematics: Applications and Interpretation HL · AHL 5: Matrices · AHL 5.5

Name Date

Systems of equations with the inverse matrix Problem set B

Write each system as a matrix equation first. Then solve it with the inverse.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. Solve each pair of equations by a matrix method.

    1. (a) \(3x + y = 9,\quad 5x + 2y = 16\)
    2. (b) \(2x + 3y = 4,\quad 4x + 5y = 9\)
    3. (c) \(x - 3y = 5,\quad 2x - 5y = 8\)

  2. Solve each system with your calculator, using the inverse matrix.

    1. (a) \(y + 2z = 7,\quad x + z = 5,\quad x + y = 3\)
    2. (b) \(2x + y - z = 5,\quad x - 3y + 2z = 4,\quad 3x + 2y + z = 13\)

  3. Find the values of \(k\) for which each system does not have a unique solution.

    1. (a) \(kx + 2y = 3,\quad 8x + ky = 1\)
    2. (b) \((k+2)x + 3y = 1,\quad 7x + (k-2)y = 5\)

  4. The equations are \(2x + 5y = 3\) and \(4x + 10y = k\).

    1. (a) Find the determinant of the matrix of coefficients.
    2. (b) Find the value of \(k\) for which there are infinitely many solutions.
    3. (c) State how many solutions there are when \(k = 9\), and what this means for the two lines.

In context

  1. At a school canteen, 3 sandwiches and 2 juices cost £11.10, and 5 sandwiches and 4 juices cost £19.30. Form a matrix equation and find the price of each item.

  2. Tomasz invests €60 000 in three funds, X, Y and Z. He puts €8000 more in Z than in X. After a year the funds have grown by 3%, 5% and 2%, and his investment has grown by €2010 in total. Form a matrix equation and find the amount in each fund.

Stretch

  1. Solve \(kx + y = 3,\quad x + ky = 2\) in terms of \(k\), where \(k \ne \pm 1\).

jmaths.xyz · AI HL · AHL 5.5 Systems of equations using the inverse matrix