mat 2401 linear algebra
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MAT 2401Linear Algebra
3.1 The Determinant of a Matrix
http://myhome.spu.edu/lauw
HW
Written Homework
Preview
How do I know a matrix is invertible?
We will look at determinant that tells us the answer.
If D=ad-bc ≠ 0 the inverse of
is given by
Recall
1 1 d bA
c aD
Therefore, if D≠0,
D is called the _________ of A
a bA
c d
If D=ad-bc = 0 the inverse of
DNE.
Fact
a bA
c d
If D=0, A is singular. To see this, for a ≠ 0, we can do the following:
1 0
0 1
11 0
0 1
a bA I
c d
b
a aad bc c
a a
R B
The Task
Given a square matrix A, we wish to associate with A a scalar det(A) that will tell us whether or not A is invertible
11 12 1
21 22 2
1 2
det( )
n
n
n n nn
a a a
a a aA
a a
A
a
Fact (3.3)
A square matrix A is invertible if and only if det(A)≠0
Interesting Comments
Interesting comments from a text: The concept of determinant is
subtle and not intuitive, and researchers had to accumulate a large body of experience before they were able to formulate a “correct” definition for this number.
n=2
11 12
21 22
11 1211 22 21 12
21 22
det( )
a aA
a a
aA
aa a a a
a a
1. Notations:
2. Mental picture for memorizing
n=3
11 12 13
21 22 23
31 32 33
11 12 13
21 22 23
31 32 33
11 22 33 11 32 23
21 12 33 21 32 13
31 12 23 31 22 13
det( )
a a a
A a a a
a a a
a a a
a a a
a a a
a a a a a a
a a a a a a
a a a a a
A
a
n=3Q1: What? Do I need to remember this?
Q2: What if A is 4x4 or bigger?
Q3: Is there a formula for 1x1 matrix?
11 12 13
21 22 23
31 32 33
11 12 13
21 22 23
31 32 33
11 22 33 11 32 23
21 12 33 21 32 13
31 12 23 31 22 13
det( )
a a a
A a a a
a a a
a a a
a a a
a a a
a a a a a a
a a a a a a
a a a a a
A
a
Observations
11 12 13
21 22 23
31 32 33
11 12 13
21 22 23
31 32 33
11 22 33 11 32 23
21 12 33 21 32 13
31 12 23 31 22 13
det( )
a a a
A a a a
a a a
a a a
a a a
a a a
a a a a a a
a a a a a a
a a a a a
A
a
Observations
11 12 13
21 22 23
31 32 33
11 12 13
21 22 23
31 32 33
11 22 33 11 32 23
21 12 33 21 32 13
31 12 23 31 22 13
det( )
a a a
A a a a
a a a
a a a
a a a
a a a
a a a a a a
a a a a a a
a a a a a
A
a
Observations
11 12 13
21 22 23
31 32 33
11 12 13
21 22 23
31 32 33
11 22 33 11 32 23
21 12 33 21 32 13
31 12 23 31 22 13
det( )
a a a
A a a a
a a a
a a a
a a a
a a a
a a a a a a
a a a a a a
a a a a a
A
a
Observations
11 12 13
21 22 23
31 32 33
11 12 13
21 22 23
31 32 33
11 22 33 11 32 23
21 12 33 21 32 13
31 12 23 31 22 13
det( )
a a a
A a a a
a a a
a a a
a a a
a a a
a a a a a a
a a a a a a
a a a a a
A
a
We need:
1. a notion of “one size smaller” but related determinants.
2. a way to assign the correct signs to these smaller determinants.
3. a way to extend the computations to nxn matrices.
Minors and Cofactors
A=[aij], a nxn Matrix.
Let Mij be the determinant of the
(n-1)x(n-1) matrix obtained from A by deleting the row and column containing aij.
Mij is called the minor of aij.
Example:
11 11
23 23
1 2 3
4 5 6
7 8 9
A
M A
M A
Minors and Cofactors
A=[aij], a nxn Matrix.
Let Cij =(-1)i+j Mij
Cij is called the cofactor of aij.
Example:
11 11
23 23
1 2 3
4 5 6
7 8 9
A
M C
M C
n=3
11 12 13
21 22 23
31 32 33
11 12 13
21 22 23
31 32 33
det( )
a a a
A a a a
a a a
a a a
A a a a
a a a
Determinants
Formally defined Inductively by using cofactors (minors) for all nxn matrices in a similar fashion.
The process is sometimes referred as Cofactors Expansion.
Cofactors Expansion (across the first column)
The determinant of a nxn matrix A=[aij] is a scalar defined by
11
11 11 21 21 1 1 1 11
11 1
if 1
detif 1
where
( 1)
n
n n k kk
kk k
a n
Aa C a C a C a C n
C M
Example 11 4 1 0
1 1 2 3
0 0 1 0
0 0 0 5
Remark
The cofactor expansion can be done across any column or any row.
1 4 1 0
1 1 2 3
0 0 1 0
0 0 0 5
Cofactors Expansion
1 1 2 21
1 1 2 21
Along the j column:
det
Along the i row:
det
where
( 1)
th
n
j j j j nj nj kj kjk
th
n
i i i i in in ik ikk
i jij ij
A a C a C a C a C
A a C a C a C a C
C M
Special Matrices and Their Determinants
(Square) Zero Matrix det(O)=?
Identity Matrixdet(I)=?
We will come back to this later….
Upper Triangular Matrix
0 for all ija i j
Upper Triangular Lower Triangular Diagonal
Lower Triangular Matrix
Upper Triangular Lower Triangular Diagonal
0 for all ija i j
Diagonal Matrix
Upper Triangular Lower Triangular Diagonal
0 for all ija i j
Diagonal Matrix
Upper Triangular Lower Triangular Diagonal
0 for all ija i j Q: T or F: A diagonal matrix is upper triangular?
Example 2
1 999 666
0 2 777
0 0 3
Determinant of a Triangular Matrix
Let A=[aij], be a nxn Triangular Matrix,
det(A)=
11
22
* * *
* * *
* *
* * nn
a
aA
a
Special Matrices and Their Determinants
(Square) Zero Matrix det(O)=
Identity Matrixdet(I)=
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