2.3 Characterizations of Invertible Matrices

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let A be an nxn matrix then the following are equivalent

  • A is an ? matrix

  • A is row equivalent to ?

  • A has ? pivots

  • The equation Ax=0 has only the ? solution

  • the columns of A are linearly ?

  • The linear transformation x→Ax is ?

  • The equation Ax=b has at least ? solution for each b in allRn

  • the columns of A ? allRn

  • the linear transformation x→Ax is ?

  • There is an nxn matrix C such that CA = ?

  • There is an nxn matrix D such that AD=?

  • AT is ?

invertible, I, n, trivial, independent, one to one, one, span, onto, I, I, invertible

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A linear transformation is invertible if there is a linear transformation S: Rn → Rn such that

S(T(x) = x and T(S(x)) = x 

S is called the inverse of T

<p>S is called the inverse of T</p>
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