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Inverse Matrix Theorem - Linear Algebra
Linear Algebra (MATH2300) w/ Prof. John Whitaker
11
Mathematics
Undergraduate 2
10/21/2012

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Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
b.
Definition
b. A is row equivalent to the n x n identity matrix.
Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
c.
Definition
c. A has n pivot positions.
Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
d.
Definition
d. The equation Ax=0 has only the trivial solution.
Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
e.
Definition
e. The columns of A form a linearly independent set.
Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
f.
Definition
f. The linear transformation x |-> Ax is one-to-one.
Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
g.
Definition
g. The equation Ax=b has at least one solution for each b in R^n.
Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
h.
Definition
h. The columns of A span R^n.
Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
i.
Definition
i. The linear transformation x |-> Ax maps R^n onto R^n.
Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
j.
Definition
j. There is an n x n matrix C such that CA = I(sub n).
Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
k.
Definition
k. There is an n x n matrix D such that AD = I(sub n).
Term
The Invertible Matrix Theorem
Let A be a square n x n matrix. Then the following statements are equivalent.
That is, for a given A, the statements are either all true or all false.
a. A is an invertible matrix.
l.
Definition
l. A^T is an invertible matrix.
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