Assignment-1 on Matrix

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Assignment Questions

  1. Elementary operations
    Explain the three basic elementary operations used in solving systems of linear equations. Perform each operation on the following matrices:
    1. \( A_1 = \begin{bmatrix}1 & 2 \\ 3 & 4 \end{bmatrix} \)
    2. \( A_2 = \begin{bmatrix}1 & 0 & 2 \\ 2 & 1 & 3 \\ 3 & 4 & 5 \end{bmatrix} \)
    3. \( A_3 = \begin{bmatrix}2 & -1 & 3 \\ 4 & 0 & 1 \\ 1 & 2 & -2 \end{bmatrix} \)
    4. \( A_4 = \begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \)
  2. Row Echelon Form:
    Reduce the following matrices to row echelon form:
    1. \( C_1 = \begin{bmatrix}1 & 2 \\ 2 & 4 \end{bmatrix} \)
    2. \( C_2 = \begin{bmatrix}1 & 2 & -1 \\ 2 & 3 & 1 \\ 1 & 1 & 0 \end{bmatrix} \)
    3. \( C_3 = \begin{bmatrix}2 & 1 & -1 \\ -3 & -1 & 2 \\ -2 & 1 & 2 \end{bmatrix} \)
  3. Reduced Row Echelon Form (RREF):
    Reduce the following matrices to RREF:
    1. \( D_1 = \begin{bmatrix}2 & 4 \\ 1 & 3 \end{bmatrix} \)
    2. \( D_2 = \begin{bmatrix}2 & 1 & -1 \\ -3 & -1 & 2 \\ -2 & 1 & 2 \end{bmatrix} \)
    3. \( D_3 = \begin{bmatrix}1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 6 & 10 \end{bmatrix} \)
  4. Rank of a Matrix:
    Find the rank of the following matrices:
    1. \( E_1 = \begin{bmatrix}1 & 2 \\ 2 & 4 \end{bmatrix} \)
    2. \( E_2 = \begin{bmatrix}1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix} \)
    3. \( E_3 = \begin{bmatrix}1 & 2 & 1 \\ 0 & 1 & -1 \\ 2 & 5 & 0 \end{bmatrix} \)
  5. Systems of Linear Equations:
    Solve the following systems using row reduction method:
    1. \( \begin{aligned} x + y &= 3, \\ 2x - y &= 0. \end{aligned} \)
    2. \( \begin{aligned} x + 2y - z &= 1, \\ 2x + y + z &= 2, \\ 3x + y &= 3. \end{aligned} \)
    3. \( \begin{aligned} x + y + z &= 6, \\ 2x - y + 3z &= 14, \\ -x + 4y + z &= 2. \end{aligned} \)
  6. Conceptual:
    Explain the relationship between the rank of a matrix and the number of solutions of a system of linear equations. Give examples for:
    1. A system with a unique solution.
    2. A system with infinitely many solutions.
    3. A system with no solution.
    4. Trivial solution.
    5. Infinitely many trivial solutions.