Matrix Problem-Solving 3: Solving Linear Systems with Matrices

Solving linear systems using matrices, showcasing Gaussian elimination, matrix inverses, and Cramer’s Rule on a blackboard.
 

Problem Set 1:

  1. Solve using Gaussian elimination:

    2x+y=54x−y=7

    Solution:

    x=3,y=−1

  2. Solve using the inverse matrix method:

    x+2y=43x−y=5

    Solution:

    x=2,y=1

  3. Apply Cramer’s Rule to solve:

    x+y=32x−y=1

    Solution:

    x=1,y=2

  4. Gaussian elimination:

    2x+y+z=9x+2y−z=53x−y+z=4

    Solution:

    x=2,y=3,z=1

  5. Inverse matrix method:

    x+3y=74x+y=10

    Solution:

    x=1,y=2

Problem Set 2:

  1. Cramer’s Rule:

    2x+y=53x−y=4

    Solution:

    x=3,y=−1

  2. Gaussian elimination:

    x+2y+3z=103x+4y−z=82x−y+z=3

    Solution:

    x=2,y=1,z=2

  3. Inverse matrix method:

    3x−y=82x+y=9

    Solution:

    x=3,y=3

  4. Gaussian elimination:

    x−y+z=52x+y+z=103x−2y−z=7

    Solution:

    x=3,y=1,z=2

  5. Cramer’s Rule:

    4x+3y=20x−y=1

    Solution:

    x=4,y=3

Problem Set 3:

  1. Gaussian elimination:

    x+y+z=62x−3y+z=−4x−y+2z=3

    Solution:

    x=3,y=1,z=2

  2. Inverse matrix method:

    5x+4y=203x−2y=5

    Solution:

    x=3,y=1

  3. Cramer’s Rule:

    2x+5y=123x+y=10

    Solution:

    x=2,y=2

  4. Gaussian elimination:

    x+y+z=72x+4y+z=153x+6y−z=10

    Solution:

    x=1,y=2,z=4

  5. Inverse matrix method:

    4x−2y=82x+3y=5

    Solution:

    x=2,y=1

Problem Set 4:

  1. Gaussian elimination:

    3x+2y−z=7x+4y+2z=102x−y+3z=15

    Solution:

    x=1,y=2,z=3

  2. Cramer’s Rule:

    5x+2y=133x−y=4

    Solution:

    x=2,y=1

  3. Inverse matrix method:

    3x+4y=182x−3y=7

    Solution:

    x=4,y=1

  4. Gaussian elimination:

    x+3y−z=22x−y+4z=123x+y−z=5

    Solution:

    x=1,y=2,z=3

  5. Cramer’s Rule:

    4x−y=9x+y=5

    Solution:

    x=2,y=3
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