3 - Determinants
3.1 The Determinant of a Matrix
ü THE DETERMINANT OF A 2¥ 2 MATRIX
Given a square matrix, we can associate it with a real number using a determinant.
Definition of the Determinant of a 2 ¥ 2 Matrix
a11 a12
The determinant of the matrix A = K
O is given by detHAL = A = a11 a22 - a12 a21 .
a21 a22
EXAMPLE 3.1.1 - Let A =
-2 3
, find A .
4 1
ü MINORS AND COFACTORS
Definition of Minors and Cofactors of a Matrix
If A is a square matrix, then the minor Mi j of the element ai j is the determinant of the matrix obtained by deleting
the ith row and jth column of A. The cofactor Ci j is given by Ci j = H-1Li+ j Mi j .
-3 4 2
EXAMPLE 3.1.2 - Let A = 6 3 1 , find the following:
4 -7 -8
a.
M11
b.
M23
c.
C11
d.
C23
2
MATH_2318_CH_03.nb
ü THE DETERMINANT OF A SQUARE MATRIX
Definition of the Determinant of a Square Matrix
If A is a square matrix (of order 2 or greater), then the determinant of A is the sum of the entries in the first row of A
multiplied by their cofactors. That is,
detHAL = A = ⁄ a1 j C1 j = a11 C11 + a12 C12 + ∫ + a1 n C1 n
n
j=1
EXAMPLE 3.1.3 - Let A =
-3 4 2
6 3 1 , find A .
4 -7 -8
Theorem 3.1 Expansion by Cofactors
Let A be a square matrix of order n. Then the determinant of A is given by
detHAL = A = ⁄ ai j Ci j = ai 1 Ci 1 + ai 2 Ci 2 + ∫ + ai n Ci n
n
j=1
or
detHAL = A = ⁄ ai j Ci j = a1 j C1 j + a2 j C2 j + ∫ + an j Cn j
n
i=1
EXAMPLE 3.1.4 - Let A =
-3 4 2
6 3 1 , find A by expanding along the second column.
4 -7 -8
MATH_2318_CH_03.nb
3
5
EXAMPLE 3.1.5 - Let A =
2
-2
2
0
0
0
4 -1
1 3
, find A .
0 6
0 7
ü TRIANGULAR MATRICES
Recall that a square matrix A is said to be upper triangular if all entries below the main diagonal are zero and it is
lower triangular if all entries above the main diagonal are zero. A matrix is said to be triangular if it is either upper
triangular or lower triangular. If a matrix is both upper triangular and lower triangular is called diagonal.
Theorem 3.2 Determinant of a Triangular Matrix
If A is a triangular matrix of order n, then its determinant is the product of the entries on the main diagonal. That is,
detHAL = A = a11 a22 ∫ an n
proof:
2
0
EXAMPLE 3.1.6 - Let A = 0
0
0
3 -1 0 8
6 0 7 1
0 -1 4 2 , find A .
0 0 3 0
0 0 0 5
3
4
MATH_2318_CH_03.nb
3.2 Determinants and Elementary Operations
ü DETERMINANTS AND ELEMENTARY ROW OPERATIONS
Theorem 3.3 Elementary Row Operations and Determinants
Let A and B be square matrices.
1.
If B is obtained from A by interchanging two rows of A, then detHBL = -detHAL.
2.
If B is obtained from A by adding a multiple of a row of A to another row of A, then detHBL = detHAL.
3.
If B is obtained from A by multiplying a row of A by a nonzero constant c, then detHBL = c detHAL.
For our purposes, the results of Theorem 3.3 can be restated as:
detHAL = -detHBL
1.
detHAL = detHBL
2.
3.
detHAL =
1
c
detHBL
1 1 1
EXAMPLE 3.2.1 - Let A = 2 -1 -2 , use elementary row operations to obtain a triangular matrix to evaluate
1 -2 -1
A .
ü DETERMINANTS AND ELEMENTARY COLUMN OPERATIONS
If we replace every appearance of the word row by the word column in Theorem 3.3, the results would still be valid.
3 -2 -1
-1
-4 2 , use elementary column operations to obtain a triangular matrix to
EXAMPLE 3.2.2 - Let A =
5 -2 -1
evaluate A .
MATH_2318_CH_03.nb
ü MATRICES AND ZERO DETERMINANTS
Theorem 3.4 Conditions That Yield a Zero Determinant
Let A is a square matrix and any of the following conditions is true, then detHAL = 0.
1.
An entire row (or an entire column) consists of zeros.
2.
Two rows (or columns) are equal.
3.
One row (or column) is a multiple of another row (or column).
proof:
2 -3 0 -4 5
-1 7 8 2 1
EXAMPLE 3.2.3 - Let A = 0 0 3 0 11 , find A .
3 2 -1 -6 7
-5 5 6 10 0
5
6
MATH_2318_CH_03.nb
3.3 Properties of Determinants
ü MATRIX PRODUCTS AND SCALAR MULTIPLES
Theorem 3.5 Determinant of a Matrix Product
If A and B are square matrices of order n, then detHA BL = detHAL detHBL.
proof:
Note that this property can be extended so that A1 A2 ∫ An = A1 ÿ A2 ∫ An .
Theorem 3.6 Determinant of a Scalar Multiple of a Matrix
If A is a square matrix of order n and c is a scalar, then the determinant of cA is detHc AL = cn detHAL.
proof:
1
0 0
3 -2 -1
1
EXAMPLE 3.3.1 - Let A = -1 -4 2 = - 3 1 0
5
2
5 -2 -1
-7 1
3
a.
Find A
b.
Find -2 A
3
-2
-1
0
14
-3
0
0
5
3
8
7
.
MATH_2318_CH_03.nb
ü DETERMINANTS AND THE INVERSE OF A MATRIX
Theorem 3.7 Determinant of an Invertible Matrix
A square matrix A is invertible (nonsingular) if and only if detHAL ∫ 0.
proof:
EXAMPLE 3.3.2 - Determine if each of the following matrices is invertible.
3 -2 -1
a.
A = -1 -4 2
5 -2 -1
b.
B=
3 -6
-2 4
If A is invertible, then detIA-1 M =
Theorem 3.8 Determinant of an Inverse Matrix
1
.
detHAL
proof:
3 -2 -1
EXAMPLE 3.3.3 - If A = -1 -4 2 , find detIA-1 M.
5 -2 -1
7
8
MATH_2318_CH_03.nb
Equivalent Conditions for a Nonsingular Matrix
If A is an n μ n matrix, then the following statements are equivalent.
1.
A is invertible.
A x = b has a unique solution for every n μ1 column matrix b.
2.
A x = 0 has only the trivial solution.
3.
4.
A is row-equivalent to In .
5.
A can be written as the product of elementary row matrices.
6.
detHAL ∫ 0
ü DETERMINANTS AND THE TRANSPOSE OF A MATRIX
Theorem 3.9 Determinant of an Inverse Matrix
If A is a square matrix, then detHAL = detIAT M.
3 -2 -1
-1
-4 2 , find detIAT M.
EXAMPLE 3.3.4 - If A =
5 -2 -1