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2
X
2 M
ATRICES
: T
HE
D
ETERMINANT
, I
DENTITY
M
ATRIX
,
AND
THE
I
NVERSE
The
identity matrix
is defined as the matrix such that
AI = IA
for all square matrices.
Therefore the product of a square matrix
A
and the identity matrix is
commutative
.
In essence, the identity matrix has no effect (same as multiply by 1).
For a matrix of order 2 by 2, the identity matrix is
=
1
0
0
1
I
.
Ex
.
Multiply.
=
×

1
0
0
1
2
4
3
1
=

×
2
4
3
1
1
0
0
1
The
determinant
is a scalar.
If a matrix has a nonzero determinant, it is said to be
non
singular
.
If the determinant is zero, the matrix is said to be
singular
.
Singularity or non
singularity is significant for some applications.
The determinant of a 2 by 2 matrix is:
bc
ad
d
c
b
a

=
det
.
Ex
.
Find the determinant of the following matrices.

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 Fall '08
 Kim
 Calculus, Determinant, Matrices

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