3
X
3 M
ATRICES
: T
HE
I
DENTITY
M
ATRIX
,
THE
D
ETERMINANT
,
AND
THE
I
NVERSE
For a three by three matrix, the
identity matrix
is
=
1
0
0
0
1
0
0
0
1
I
.
It has the same properties
as a two by two matrix, in that for a three by three matrix A,
A
IA
AI
+
=
.
The
determinant
of a three by three matrix is defined in terms of the determinant of two by
two submatrices:
+

=
f
e
c
b
g
i
h
c
b
d
i
h
f
e
a
i
h
g
f
e
d
c
b
a
To find the determinant of the 3 x 3 matrix, find the determinant of each of the 2 x 2 matrices.
The matrices are composed of the elements in the first column multiplied by the 2 x 2 matrix
formed by the four elements not in that column or row.
Since
a
is in the first row and the first column, this is an even sum and
a
is positive.
Since
d
is in the second row and the first column, this is an odd sum and
d
is negative.
Lastly, since
g
is in the third row and the first column, this is an even sum and
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 Fall '08
 Kim
 Calculus, Linear Algebra, Determinant, Matrices, Invertible matrix, Identity matrix

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