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(Section 8.3: The Inverse of a Square Matrix)
8.47
SECTION 8.3: THE INVERSE OF A SQUARE MATRIX
PART A: (REVIEW) THE INVERSE OF A REAL NUMBER
If
a
is a nonzero real number, then
aa
−
1
=
a
1
a
⎛
⎝
⎜
⎞
⎠
⎟
=
1
.
a
−
1
, or
1
a
, is the multiplicative inverse of
a
, because its product with
a
is 1,
the multiplicative identity.
Example
3
1
3
⎛
⎝
⎜
⎞
⎠
⎟
=
1
, so 3 and
1
3
are multiplicative inverses of each other.
PART B: THE INVERSE OF A SQUARE MATRIX
If
A
is a square
n
×
n
matrix, sometimes there exists a matrix
A
−
1
(“
A
inverse”) such that
AA
−
1
=
I
n
and
A
−
1
A
=
I
n
.
An invertible matrix and its inverse commute with respect to matrix multiplication.
Then,
A
is invertible (or nonsingular
), and
A
−
1
is unique.
In this course
, an invertible matrix is assumed to be square.
Technical Note: A nonsquare matrix may have a left inverse matrix or a right
inverse matrix that “works” on one side of the product and produces an identity
matrix. They cannot be the same matrix, however.
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8.48
PART C:
FINDING
A

1
We will discuss a shortcut for
2
×
2
matrices in
Part F
.
Assume that
A
is a given
n
×
n
(square) matrix.
A
is invertible
⇔
Its RRE Form is the identity matrix
I
n
(or simply
I
).
It turns out that a sequence of EROs that takes you from an invertible matrix
A
down to
I
will also take you from
I
down to
A
−
1
. (A good Linear Algebra
book will have a proof for this.) We can use this fact to efficiently find
A
−
1
.
We construct
A
I
⎡
⎣
⎤
⎦
. We say that
A
is in the “left square” of this matrix,
and
I
is in the “right square.”
We apply EROs to
A
I
⎡
⎣
⎤
⎦
until we obtain the RRE Form
I
A
−
1
⎡
⎣
⎤
⎦
.
That is, as soon as you obtain
I
in the left square, you grab the matrix in the
right square as your
A
−
1
.
If you
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 Fall '11
 staff
 Calculus

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