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Gauss-Jordan Elimination Method: Finding Matrix Inverse and Basis, Quizzes of Linear Algebra

University of Illinois - ChicagoLinear Algebra

Solutions to two problems in math 310. The first problem involves using the gauss-jordan elimination method to find the inverse of a given matrix. The second problem examines whether the vectors (1, 2, −2)t, (1, 1, 2)t, and (3, −1, 1)t form a basis of r3.

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2011/2012

Uploaded on 05/18/2012

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MATH 310
Self-quiz 16
1. Use the Gauss-Jordan elimination method to find the inverse of the
matrix:
1 2 −1
1 1 1
2−1 2
2. Examine whether the vectors (1,2,−2)T, (1,1,2)Tand (3,−1,1)Tform
a basis of R3or not.
pf3
pf4
pf5

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MATH 310

Self-quiz 16

  1. Use the Gauss-Jordan elimination method to find the inverse of the matrix:

 

  1. Examine whether the vectors (1, 2 , −2)T^ , (1, 1 , 2)T and (3, − 1 , 1)T^ form a basis of R^3 or not.

MATH 310

Self-quiz 16

  1. Use the Gauss-Jordan elimination method to find the inverse of the matrix:

 

Solution: We will apply the Gauss-Jordan elimination method to the ma- trix:

 

till the matrix on the left becomes the identity matrix.

We have:

 

−R 1 + R 2

− 2 R 1 + R 3

−R 2

5 R 2 + R 3

λ + μ + 3ν = 0 2 λ + μ − ν = 0 − 2 λ + 2μ + ν = 0

We will solve this system by applying the Gauss-Jordan elimination method to the augmented matrix of the system:

 

− 2 R 1 + R 2

2 R 1 + R 3

−R 2

− 4 R 2 + R 3

− 211 R 3

We can now return to our unknowns:

λ + μ + 3ν = 0 μ + 7ν = 0 ν = 0

which easily gives λ = μ = ν = 0.

Therefore the three vectors are linearly independent and form a basis of R^3.