Monday March 16
−
Lecture 28 :
Determinants and
invertibility
.
(Refers to sections
5.2)
Objectives
:
1.
Recognize that det(
AB
) = det(
A
)det(
B
).
2.
Recognize that det(
cA
) =
c
n
det(
A
).
3.
Recognize that a matrix A is invertible iff det(
A
)
≠
0.
4.
Recognize that det(
A
1
) = 1 / det(
A
).
5.
Define the
adjoint
of a matrix.
6.
Find the inverse of a matrix by using the
adjoint
of the matrix.
7.
Apply
Cramers
rule to find to solve a system of linear equations.
28.1
Theorem
−
If
A
and
B
are two square matrices of same dimension then
det(
AB
) =
det(
A
)det(
B
)
.
(Proof is omitted)
28.2
Theorem
−
If
A
is an
n
×
n
matrix and
c
is a scalar, then the
det(
cA
) =
c
n
det(
A
)
.
28.3
Theorem.
1.
If the square matrix
A
has an inverse then det(
A
)
≠
0
.
2.
If
A
is a matrix such that det(
A
)
≠
0 then
A
is invertible.
Proof of
"
If the square matrix A has an inverse then
det(
A
)
≠
0":
o
Suppose that
A
is invertible.
o
Then
A
is a product of elementary matrices,
A
=
E
1
E
2
E
3
...
E
n
.
o
Then det(
A
) = det(
E
1
)det(
E
2
)det(
E
3
)...det(
E
n
). Why?
o
The determinant of an elementary matrix is either
−
1, a nonzero scalar or 1,
depending on whether it is an elementary matrix of type I, II or III,
respectively.
o
The determinant of an elementary matrix is never zero. Why?
o
Thus det(
A
)
≠
0.
Proof of
"
If A is a matrix such that
det(
A
)
≠
0
then A is invertible
":
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o
Suppose now that det(
A
)
≠
0.
o
Then det(
A
RREF
) is not zero. Why? (det(
A
RREF
) is the product of det(
A
) and the
determinant of elementary matrices.)
o
Thus
A
RREF
does not have a row or zeros.
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 Winter '08
 All
 Linear Algebra, Determinant, Invertible matrix, Det

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