MAT 2401
Linear 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.
Recall
If D=ad-bc ≠ 0 the
inverse of
a b 
A

c
d


is given by
1  d b 
A  
D  c a 
1
Therefore, if D≠0,
D is called the _________ of A
Fact
If D=ad-bc = 0 the
inverse of
a b 
A

c
d


DNE.
If D=0, A is singular.
To see this, for a ≠ 0, we can
do the following:
a b
 A I   c d

1 0


0 1
b

1

a

 0 ad  bc

a
  R B
1
a
c

a

0

1

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
 a11 a12
a
a22
21

A


 an1 an 2
a1n 
a2 n 


ann 
det( 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
a12 
a
A   11

a
a
22 
 21
a11 a12
det( A) 
 a11a22  a21a12
a21 a22
1. Notations:
2. Mental picture for memorizing
n=3
 a11 a12 a13 
A   a21 a22 a23 
 a31 a32 a33 
a11 a12 a13
det( A)  a21
a22
a23
a31
a32
a33
 a11a22 a33  a11a32 a23
 a21a12 a33  a21a32 a13
 a31a12 a23  a31a22 a13
n=3
 a11 a12 a13 
A   a21 a22 a23 
 a31 a32 a33 
a11 a12 a13
det( A)  a21
a22
a23
a31
a32
a33
 a11a22 a33  a11a32 a23
 a21a12 a33  a21a32 a13
 a31a12 a23  a31a22 a13
Q1: What? Do I need to
remember this?
Q2: What if A is 4x4 or bigger?
Q3: Is there a formula for 1x1
matrix?
Observations
 a11 a12 a13 
A   a21 a22 a23 
 a31 a32 a33 
a11 a12 a13
det( A)  a21
a22
a23
a31
a32
a33
 a11a22 a33  a11a32 a23
 a21a12 a33  a21a32 a13
 a31a12 a23  a31a22 a13
Observations
 a11 a12 a13 
A   a21 a22 a23 
 a31 a32 a33 
a11 a12 a13
det( A)  a21
a22
a23
a31
a32
a33
 a11a22 a33  a11a32 a23
 a21a12 a33  a21a32 a13
 a31a12 a23  a31a22 a13
Observations
 a11 a12 a13 
A   a21 a22 a23 
 a31 a32 a33 
a11 a12 a13
det( A)  a21
a22
a23
a31
a32
a33
 a11a22 a33  a11a32 a23
 a21a12 a33  a21a32 a13
 a31a12 a23  a31a22 a13
Observations
 a11 a12 a13 
A   a21 a22 a23 
 a31 a32 a33 
a11 a12 a13
det( A)  a21
a22
a23
a31
a32
a33
 a11a22 a33  a11a32 a23
 a21a12 a33  a21a32 a13
 a31a12 a23  a31a22 a13
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:
1 2 3
A   4 5 6 
7 8 9 
M 11 
A11 
M 23 
A23 
Minors and Cofactors
A=[aij], a nxn Matrix.
Let Cij =(-1)i+j Mij
Cij is called the cofactor
of aij.
Example:
1 2 3
A   4 5 6 
7 8 9 
M 11 
C11 
M 23 
C23 
n=3
 a11 a12 a13 
A   a21 a22 a23 
 a31 a32 a33 
a11 a12 a13
det( A)  a21 a22 a23 
a31 a32 a33
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
a11
if n  1


n
det A  
a11C11  a21C21   an1Cn1   ak1Ck1 if n  1
k 1

where
Ck1  ( 1) k 1 M k1
Example 1
1 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
Along the jth column:
det A  a1 j C1 j  a2 j C2 j 
n
 anj Cnj   akj Ckj
k 1
Along the i th row:
det A  ai1Ci1  ai 2Ci 2 
where
Cij  (1)i  j M ij
n
 ainCin   aik Cik
k 1
Special Matrices and Their
Determinants


(Square) Zero Matrix
det(O)=?
Identity Matrix
det(I)=?
We will come back to this later….
Upper Triangular Matrix
aij  0 for all i  j














Upper Triangular














Lower Triangular














Diagonal
Lower Triangular Matrix
aij  0 for all i  j














Upper Triangular














Lower Triangular














Diagonal
Diagonal Matrix
aij  0 for all i  j














Upper Triangular














Lower Triangular














Diagonal
Diagonal Matrix
Q: T or F: A diagonal matrix
is upper triangular?
aij  0 for all i  j














Upper Triangular





















Lower Triangular







Diagonal
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)=
 a11 *
* a
22
A
*
*

*
*
*
*
* 
* 


ann 
Special Matrices and Their
Determinants


(Square) Zero Matrix
det(O)=
Identity Matrix
det(I)=