2.3 Characterizations of Invertible Matrices

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2.3 Characterizations of Invertible Matrices
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REVIEW
Invertible (Nonsingular)
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REVIEW
Theorem 7


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REVIEW
How do you find the inverse of a matrix, A?
If A is a 2x2 matrix
In general,

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Section 2.3 is a review of most of the topics in
Chapter 1, in relation to systems of n linear
equations in n unknowns and to square matrices.
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• Theorem 8 (The Invertible Matrix Theorem)
• Let A be a square matrix.
• Then the following statements are equivalent
• A is an invertible matrix.
• A is row equivalent to the identity
  matrix.
• A has n pivot positions.
• The equation Ax0 has only the trivial solution.
• The columns of A form a linearly independent set.
• The linear transformation x Ax is
  one-to-one.
• The equation Axb has at least one solution for
  each b in .
• The columns of A span .
• The linear transformation x Ax maps
  onto .
• There is an matrix C such that CAI.
• There is an matrix D such that ADI.
• is an invertible matrix.