The invertible matrix theorem

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Intros
Lessons
  1. Characterizations of Invertible Matrices Overview:
  2. 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
  3. How to apply the Invertible Matrix Theorem
    • Showing a Matrix is invertible
    • Shortcuts to know certain statements
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Examples
Lessons
  1. Showing a Matrix is invertible or not invertible
    Is the following matrix invertible?
    determine whether the matrix is invertible
    1. Is the following matrix invertible? Use as few calculations as possible.
      determine whether the matrix is invertible
      1. Understanding the Theorem
        Assume that AA is a square n×nn \times n matrix. Determine if the following statements are true or false:
        1. If AA is an invertible matrix, then the linear transformation xx Ax Ax maps Rn\Bbb{R}^n onto Rn\Bbb{R}^n.
        2. 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
        3. If the equation Ax=0Ax=0 has only the trivial solution, then AA is not invertible.
        4. If the equation Ax=0Ax=0 has a non-trivial solution, then AA has less than nn pivots.
      2. Can a square matrix with two identical rows be invertible? Why or why not?
        1. Let AA and BB be n×nn \times n matrix. Show that if ABAB is invertible, so is BB.