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The Timpson Lyceum · Linear Algebra (2026)

1.1 Exercises

Problem sheet · 20 points · 4 extra credit

Systems of Linear Equations

  1. 1. (2 pts) Solve the system by elimination or substitution:

    x1+2x2=43x1−x2=5.\begin{aligned} x_1 + 2x_2 &= 4 \\ 3x_1 - x_2 &= 5. \end{aligned}

  2. 2. (3 pts) Write the augmented matrix of the following system and use elementary row operations to solve it:

    x1+x2+x3=62x1−x2+x3=3x1+2x2−x3=2.\begin{aligned} x_1 + x_2 + x_3 &= 6 \\ 2x_1 - x_2 + x_3 &= 3 \\ x_1 + 2x_2 - x_3 &= 2. \end{aligned}

  3. 3. (3 pts) Consider the system

    x1−3x2=12x1+h x2=4.\begin{aligned} x_1 - 3x_2 &= 1 \\ 2x_1 + h\,x_2 &= 4. \end{aligned}

    Find all values of hh for which the system has

    (a) a unique solution,

    (b) no solution,

    (c) infinitely many solutions.

  4. 4. (3 pts) Each matrix below is the augmented matrix of a linear system that has already been row reduced. In each case state whether the system is consistent, and if so how many solutions it has.

    (a) [1003010−10012]\left[\begin{array}{ccc|c} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 2 \end{array}\right]

    (b) [1−20400150000]\left[\begin{array}{ccc|c} 1 & -2 & 0 & 4 \\ 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right]

    (c) [103201−140006]\left[\begin{array}{ccc|c} 1 & 0 & 3 & 2 \\ 0 & 1 & -1 & 4 \\ 0 & 0 & 0 & 6 \end{array}\right]

  5. 5. (3 pts) State whether each of the following is true or false, and justify each answer.

    (a) Every elementary row operation is reversible.

    (b) A system of two equations in three unknowns can have exactly one solution.

    (c) If a linear system is inconsistent, then the reduced form of its augmented matrix has a row of the form [ 0    0    ⋯    0∣b ][\,0 \;\; 0 \;\; \cdots \;\; 0 \mid b\,] with b≠0b \neq 0.

  6. 6. (3 pts) Explain why performing any one elementary row operation on the augmented matrix of a linear system does not change its solution set. Address all three types of operation (replacement, interchange, scaling).

  7. 7. (3 pts) Describe geometrically the solution set of each system in R2\mathbb{R}^2, and state how many solutions it has.

    (a) x1−x2=12x1−2x2=5\begin{aligned} x_1 - x_2 &= 1 \\ 2x_1 - 2x_2 &= 5 \end{aligned}

    (b) x1−x2=12x1−2x2=2\begin{aligned} x_1 - x_2 &= 1 \\ 2x_1 - 2x_2 &= 2 \end{aligned}

  8. 8. (4 pts ••) Suppose u=(u1,…,un)u = (u_1, \dots, u_n) and v=(v1,…,vn)v = (v_1, \dots, v_n) are two distinct solutions of a system of linear equations. For a real number tt, let ww be the tuple whose kk-th entry is wk=uk+t(uk−vk)w_k = u_k + t(u_k - v_k). Prove that ww is also a solution for every tt, and conclude that the system has infinitely many solutions.