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Matrix Operations

Part 5: Systems of Linear Equations

  1. Use your helper application to solve the system
  2. x1 - x3 + 3x4 = 1
    2x1 + x2 + 4x3 - 2x4 = -2
    - 5x2 + x4 = 11
    -x1 + 2x2 - x3 + 3x4 = 3

    Note that the matrix of coefficients for this system is the matrix A already defined in your worksheet. Thus, the system can be written in matrix form as Ax = b, where b is the column vector [1,-2,11,3]T. Check the answer by multiplying the column vector x = [x1,x2,x3,x4]T on the left by A to see if the result is b.

  3. Recall that A is invertible. Calculate x as A-1b to see if you get the same result. Note that this system has a unique solution.
  4. Now solve the system
  5. 2x1 + x2 + 7x4 = 1
    -2x1 + 5x2 + x3 + 2x4 = -2
    4x1 + x2 + 3x3 - 6x4 = 11
    4x1 - 4x2 - x3 + 5x4 = 3


    What does the answer mean? The matrix of coefficients for this system is B, which we saw was not invertible. This is an example of a linear system with infinitely many solutions.

  6. Write down one specific solution x of the system in the preceding step. Multiply on the left by B, and confirm that Bx really is b. Now do it again with a different solution.
  7. Change the constant term in the first equation to anything other than 1, and try to solve the resulting system. What happens? This is an example of an inconsistent system -- it has no solution.

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