The invertible matrix theorem

The invertible matrix theorem

Lessons

The Invertible Matrix Theorem states the following:
Let AA be a square n×nn \times n matrix. Then the following statements are equivalent. That is, for a given AA, the statements are either all true or all false.
1. AA is an invertible matrix.
2. AA is row equivalent to the n×nn \times n identity matrix.
3. AA has nn pivot positions.
4. The equation Ax=0Ax=0 has only the trivial solution.
5. The columns of AA form a linearly independent set.
6. The equation Ax=bAx=b has at least one solution for each bb in Rn\Bbb{R}^n.
7. The columns of AA span Rn\Bbb{R}^n.
8. The linear transformation xx Ax Ax maps Rn\Bbb{R}^n onto Rn\Bbb{R}^n.
9. There is an n×nn \times n matrix CC such that CA=ICA=I.
10. There is an n×nn \times n matrix DD such that AD=IAD=I.

There are extensions of the invertible matrix theorem, but these are what we need to know for now. Keep in mind that this only works for square matrices.
  • Introduction
    Characterizations of Invertible Matrices Overview:
    a)
    The Invertible Matrix Theorem
    • only works for n×nn \times n square matrices
    • If one is true, then they are all true
    • If one is false, then they are all false

    b)
    How to apply the Invertible Matrix Theorem
    • Showing a Matrix is invertible
    • Shortcuts to know certain statements


  • 1.
    Showing a Matrix is invertible or not invertible
    Is the following matrix invertible?
    determine whether the matrix is invertible

  • 2.
    Is the following matrix invertible? Use as few calculations as possible.
    determine whether the matrix is invertible

  • 3.
    Understanding the Theorem
    Assume that AA is a square n×nn \times n matrix. Determine if the following statements are true or false:
    a)
    If AA is an invertible matrix, then the linear transformation xx Ax Ax maps Rn\Bbb{R}^n onto Rn\Bbb{R}^n.

    b)
    If there is an n×nn \times n matrix CC such that CA=ICA=I, then there is an n×nn \times n matrix DD such that AD=IAD=I

    c)
    If the equation Ax=0Ax=0 has only the trivial solution, then AA is not invertible.

    d)
    If the equation Ax=0Ax=0 has a non-trivial solution, then AA has less than nn pivots.


  • 4.
    Can a square matrix with two identical rows be invertible? Why or why not?

  • 5.
    Let AA and BB be n×nn \times n matrix. Show that if ABAB is invertible, so is BB.