Invertible Matrix Theorem
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11 Terms
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Assumptions for A, a square n × n invertible matrix
Invertible Matrix Theorem
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A is row equivalent to
the n × n identity matrix.
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A has n amount of ___
pivot positions
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The equation Ax = 0 has ____ solution.
only the trivial
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(v) The columns of A form a ____ set.
linearly independent
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The linear transformation x ↦ Ax
is one-to-one.
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The equation Ax = b has at least
one solution for each b in Rn
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What can be said about the span of A?
The columns of A span Rn.
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The linear transformation x ↦ Ax maps
Rn onto Rn.
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What other matrices exist?
There is an n × n matrix C such that CA = I.
There is an n × n matrix D such that AD = I.
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What can be said about the transpose of A?
A^T is an invertible matrix.