2.3 Characterization of Invertible Matrices
Theorem 8 pulls facts together. Memorize it.
Theorem 2.8 (The Invertible Matrix Theorem):
Let A be a square n x n matrix. The following
statements are equivalent.
a. A is an invertible matrix.
b. A is row equivalent to In.
c. A has n pivot positions.
d. The equation Ax = 0 has only the trivial
solution.
e. The columns of A form a linearly
independent set.
f. The linear transformation x → Ax is one-toone.
g. The equation Ax = b has at least one
solution for each b in Rn.
h. The columns A span Rn.
i. The linear transformation x → Ax maps Rn
onto Rn.
j. There is an n x n matrix C s.t. CA = In.
k. There is an n x n matrix D s.t. AD = In.
l. AT is an invertible matrix.
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Example: We can use the IMT to determine
whether a matrix in invertible.
Let
 1  3 0
A  - 4 11 1
 2 7 3
Use row operations to get information about
the pivot positions: R2 + 4R1 and R3 + (-2R1)
1  3 0
A  0  1 1
0 1 3
R3 + R2
1  3 0
A  0  1 1
0 0 4 A is a 3x3 matrix with has
3 pivot positions, so A in invertible.
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Ex: Suppose H is a 5 x 5 matrix, and suppose
v is a vector in R5 which is not a linear
combination of the columns of H. What can
you say about the number of solutions to
Hx = 0?
Answer: By the definition of span, since
v  R5 and is not a linear combination of the
columns of H, the columns of H do not span
R5. Thus, part h of the IMT fails, so part d
also fails, and Hx = 0 has more than just the
trivial solution, hence infinitely many
solutions.
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