3. det ( A ) is a real number (can be negative also)
det ( A ) is also written as
Only a square n x n matrix has a determinant
det ( AB ) = det ( A ) det ( B )
A matrix A is not invertible (singular matrix) if
and only if det ( A ) = 0
A matrix A is invertible (nonsingular matrix) if
and only if det ( A ) ≠ 0
3.6 Determinants : Definition
A
4. 1x1 matrix : A = [ a11 ], then det (A) = a11
2x2 matrix :
det (A) = a11 a22 - a12 a21
11 12
21 22
a a
a a
A
3.6 Determinants : Examples
5. 3x3 matrix
det (A) = a11 a22 a33 + a12 a23 a31 + a13 a21 a32
-a11 a23 a32 - a12 a21 a33 -a13 a22 a31
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
A
Note: This cannot be extended to higher order matrices.
3.6 Determinants : Examples
6. 3x3 matrix
minus
det (A) =
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
A
3.6 Determinants : Examples
11 12 13
22 23 21 23 21 22
32 33 31 33 31 32
det det deta a a
a a a a a a
a a a a a a
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
8. Defn - Let A = [ aij ] be an n x n matrix.
Let Mpq be the (n – 1) x (n – 1) submatrix of A
obtained by deleting the pth row and qth column of
A. The determinant det(Mpq) is called the minor
of apq
Defn - Let A = [ aij ] be an n x n matrix.
The cofactor Apq of apq is defined as
Apq = (–1)p+q det(Mpq)
3.6 Determinants : Cofactor Expansion
9. 1 2 3
4 5 6
7 8 9
A
2
11 11 11
3
12 12 12
4
31 31 31
5 6
1 det 3
8 9
4 6
1 det 6
7 9
2 3
1 det 3
5 6
A
A
A
M M
M M
M M
3.6 Determinants : Cofactor Expansion - Example
11. Theorem - Let A = [ aij ] be an n x n matrix. Then
det(A) = ai1Ai1+ ai2Ai2 + L + ain Ain
(expansion of det(A) with respect to row i )
det(A) = a1j A1j+ a2j A2j + L + anj Anj
(expansion of det(A) with respect to column j )
det(A) can be calculated by expansion along the
row (or column) of our choice.
3.6 Determinants : Cofactor Expansion
12. Expand the determinant of a 3 x 3 matrix with
respect to first row 11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
A
1 1 22 23
11 22 33 23 32
32 33
1 2 21 23
12 21 33 23 31
31 33
1 3 21 22
13 21 32 22 31
31 32
1
1
1
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
So det(A) = a11A11 + a12A12 + a13A13
3.6 Determinants : Cofactor Expansion
13. Expand the determinant of a 3 x 3 matrix with
respect to first column of A
1 1 22 23
11 22 33 23 32
32 33
2 1 12 13
21 13 32 12 33
32 33
3 1 12 13
31 12 23 13 22
22 23
1
1
1
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
So det(A) = a11A11 + a21A21 + a31A31
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
A
3.6 Determinants : Cofactor Expansion
14. Evaluate
Pick row or column with large number of
zeros, such as column 2
1 2 3 4
4 2 1 3
det
3 0 0 3
2 0 2 3
A
3.6 Determinants : Cofactor Expansion - Example
17. Theorem - Let A be a square matrix.
Then det ( A ) = det ( AT )
Since det ( A ) = det ( AT ), the properties of
determinants with respect to row manipulations are true
also for the corresponding column manipulations
Theorem - If a row (column) of A consists entirely of
zeros, then det ( A ) = 0.
Proof - Let the i th row of A be all zeros. Each term of
det ( A ) contains a factor from the i th row of A. So,
det ( A ) = 0.
3.6 Determinants : Properties
18. Theorem – Interchange two rows (columns) of A
If matrix B is obtained from matrix A by interchanging
two rows (columns) of A,
then det (B) = - det (A)
Theorem - If two rows (columns) of A are equal, then
det (A) = 0.
Proof - Suppose rows r and s of A are equal.
Interchange rows r and s to obtain matrix B. Then
det (B) = - det (A). However, B = A so
det (B) = det (A). Thus det (A) = 0.
3.6 Determinants : Properties
19. Theorem – Multiply row (column) i of A by c ≠ 0.
If B is obtained from A by multiplying a row (column) of
A by a real number c, then
det ( B ) = c det ( A ).
Proof - Suppose the ith row of A is multiplied by c to
get B. Each term of det ( B ) contains a single factor
from the ith row of B. Thus, each term of det ( B )
consists of a single factor of c times the corresponding
term of det ( A ). So, det ( B ) = c det ( A ).
3.6 Determinants : Properties
20. Theorem – Add c times row (column) r of A to row
(column) s of A, r ≠ s
If B = [ bij ] is obtained from A = [ aij ] by adding to each
element of rth row (column) of A, c times the
corresponding element of the sth row (column), r ≠ s, of
A, then
det(B) = det(A)
3.6 Determinants : Properties
21. Theorem - Let A = [ aij ] be an upper (lower) triangular
matrix, then
det(A) = a11a22 … ann.
The determinant of a triangular matrix is the product of the
elements on the main diagonal.
3.6 Determinants : Properties
22. Theorem - If A is an nxn matrix, then Ax = 0
has a nontrivial solution if and only if det(A) = 0.
Proof - Ax = 0 has a nontrivial solution if and only if A
is singular. If A is singular, then det(A) = 0.
Theorem - If A is an nxn matrix, then Ax = b
has a unique solution if and only if det(A) 0.
3.6 Determinants : Properties
≠
23. Theorem - If A and B are n x n matrices, then
det(AB) = det(A) det(B)
Proof - If A or B is singular, then AB is singular and the result
follows immediately. So, suppose that A and B are both
nonsingular. Then A and B can be expressed as the product of
elementary matrices.
A = E1E2 L Ep and B = F1F2 L Fq
det( A ) = det( E1 ) det( E2 ) L det( Ep )
det( B ) = det( F1 ) det( F2 ) L det( Fq )
det( AB ) = det( E1E2 L Ep F1F2 L Fq ) =
det( E1 ) det( E2 ) L det( Ep ) det( F1 ) det( F2 ) L det( Fq ) =
det( A ) det( B )
3.6 Determinants : Properties
24. Theorem - If A is nonsingular, then
det(A–1) = 1/det(A)
Proof - Let In be the nxn identity. Then AA–1 = In
and det(In) = 1.
By the preceding theorem, det(A) det(A–1) = 1,
so det(A–1) = 1/det(A)
3.6 Determinants : Properties
