The fundamental theorem of linear algebra states If A is a n x n matrix, then the following statements are equivalent 1.
Posted: Tue Jul 12, 2022 12:41 pm
The fundamental theorem of linear algebra states
If A is a n x n matrix, then the following statements areequivalent
1. det(A) does not equal 0
2. A can be written as the product of elementary matrices
3. A is row equivalent to In
4. Ax = 0 has only the trivial solution
5. Ax = b has a unique solution for every n x 1 column matrixb
6. A is invertible
Convince yourself why conditions 2 and 3 must be true. To dothis, pick two 3 x 3 matrices: one that is invertible and one thatis not. Do not use the identity matrix or an elementarymatrix. Set up the homogeneous andnon-homogenous system represented by the matrices.Solve the augmented system of equationsusing the elementary row operations on matrices. Upon completion,each matrix should be in reduced row echelon form. Show theresulting 4 augmented matrices. Explain what the solutions are toeach of the four systems. Then explain why conditions two and threemust be true.
If A is a n x n matrix, then the following statements areequivalent
1. det(A) does not equal 0
2. A can be written as the product of elementary matrices
3. A is row equivalent to In
4. Ax = 0 has only the trivial solution
5. Ax = b has a unique solution for every n x 1 column matrixb
6. A is invertible
Convince yourself why conditions 2 and 3 must be true. To dothis, pick two 3 x 3 matrices: one that is invertible and one thatis not. Do not use the identity matrix or an elementarymatrix. Set up the homogeneous andnon-homogenous system represented by the matrices.Solve the augmented system of equationsusing the elementary row operations on matrices. Upon completion,each matrix should be in reduced row echelon form. Show theresulting 4 augmented matrices. Explain what the solutions are toeach of the four systems. Then explain why conditions two and threemust be true.