STK 500 Pengantar Teori Statistika Determinant of Matrix
Transcript of STK 500 Pengantar Teori Statistika Determinant of Matrix
STK 500
Pengantar Teori Statistika
Determinant of Matrix
The Determinant of a Matrix
The determinant of a matrix A is commonly denoted by |A| or det A.
Determinants exist only for square matrices.
They are a matrix characteristic (that can be somewhat tedious to compute).
The Determinant for a 2x2 Matrix
If we have a matrix A such that
then
For example, the determinant of
is
Determinants for 2x2 matrices are easy!
11 22 12 21= a a - a aA
11 12
21 22
a aa a
A
1 23 4
A
11 22 12 211 2= = a a - a a 1 4 - 2 3 = -23 4
A
2221
1211
aa
aaA
The Determinant for a 3x3 Matrix
If we have a matrix A such that
Then the determinant is
which can be expanded and rewritten as
11 12 13
21 22 23
31 32 33
a a aa a aa a a
A
22 23 21 23 21 2211 12 13
31 3232 33 31 33
a a a a a adet = = a - a +a
a aa a a aA A
11 1122 33 23 32 12 23 31
12 21 33 13 21 32 13 22 31
det = = a a a - a a a + a a a
- a a a + a a a - a a a
A A(Why?)
The Determinant for a 3x3 Matrix
If we rewrite the determinants for each of
the 2x2 submatrices in
22 23 21 23 21 2211 12 13
31 3232 33 31 33
a a a a a adet = = a - a +a
a aa a a aA A
A 11 1122 33 23 32 12 23 31 12 21 33 13 21 32 13 22 31= a a a - a a a + a a a - a a a + a a a - a a a
as 22 2322 33 23 32
32 33
21 2321 33 23 31
31 33
21 2221 32 22 31
31 32
a a=a a - a a ,
a a
a a=a a - a a , and
a a
a a=a a - a a
a a
by substitution we have
The Determinant for a 3x3 Matrix
Note that if we have a matrix A such that
Then |A| can also be written as
or
or
11 12 13
21 22 23
31 32 33
a a aa a aa a a
A
22 23 21 23 21 2211 12 13
31 3232 33 31 33
a a a a a adet = = a - a +a
a aa a a aA A
12 13 11 23 11 1221 22 23
31 3232 33 31 33
a a a a a adet = = -a +a - a
a aa a a aA A
12 13 11 13 11 1231 32 33
22 23 21 23 21 22
a a a a a adet = = a - a +a
a a a a a aA A
The Determinant for a 3x3 Matrix
To do so first create a matrix of the same dimensions as A consisting only of alternating signs (+,-,+,…)
+ - +- + -+ - +
The Determinant for a 3x3 Matrix
Then expand on any row or column (i.e., multiply each element in the selected row/column by the corresponding sign, then multiply each of these results by the determinant of the submatrix that results from elimination of the row and column to which the element belongs
For example, let’s expand on the second column
11 12 13
21 22 23
31 32 33
a a aa a aa a a
A
The Determinant for a 3x3 Matrix
The three elements on which our expansion is based will be a12, a22, and a32. The corresponding signs are -, +, -.
+ - +- + -+ - +
The Determinant for a 3x3 Matrix
So for the first term of our expansion we will multiply -a12 by the determinant of the matrix formed when row 1 and column 2 are eliminated from A (called the minor and often denoted Arc where r and c are the deleted rows and columns):
11 12 1321 23
21 22 23 1231 33
31 32 33
a a aa a
a a a so a a
a a aA A
which gives us 21 23
1231 33
a a-a
a a
This product is called a cofactor.
The Determinant for a 3x3 Matrix
For the second term of our expansion we will multiply a22 by the determinant of the matrix formed when row 2 and column 2 are eliminated from A:
11 12 1311 13
21 22 23 2231 33
31 32 33
a a aa a
a a a so a a
a a aA A
which gives us
11 1322
31 33
a aa
a a
The Determinant for a 3x3 Matrix
Finally, for the third term of our expansion we will multiply -a32 by the determinant of the matrix formed when row 3 and column 2 are eliminated from A:
11 12 1311 13
21 22 23 2221 23
31 32 33
a a aa a
a a a so a a
a a aA A
which gives us 11 13
3221 23
a a-a
a a
The Determinant for a 3x3 Matrix
Putting this all together yields
So there are nine distinct ways to calculate the determinant of a 3x3 matrix!
A A 21 23 11 13 11 1312 22 32
31 33 31 33 21 23
a a a a a adet = = -a +a - a
a a a a a a
The Determinant Laplace formula
Theorem (Determinant as a Laplace expansion)
Suppose A = [aij] is an nxn matrix and i,j= {1, 2, ...,n}. Then the determinant
m n
i+j i+j
ij ij ij ijj=1 i=1
det = = a -1 = a -1A A A A
Note that this is referred to as the method of cofactors and can be used to find the determinant of any square matrix.
14 Pierre-Simon Laplace (1749–1827).
The Determinant for a 3x3 Matrix – An Example
Suppose we have the following matrix A:
Using row 1 (i.e., i=1), the determinant is:
m
1+j
1j 1jj=1
det = = a -1 1(2) 2( 8) 3( 11) 15A A A
Note that this is the same result we would achieve using any other row or column!
1 2 3= 2 5 4
1 -3 -2A
For ONLY a 3x3 matrix write down the first two columns after the third column
3231
2221
1211
333231
232221
131211
aa
aa
aa
aaa
aaa
aaa
Sum of products along red arrow minus sum of products along blue arrow
This technique works only for 3x3 matrices
332112322311312213
322113312312332211
aaaaaaaaa
aaaaaaaaa)A
det(
aaa
aaa
aaa
333231
232221
131211
A
The Determinant for a 3x3 Matrix
21-24013-42
A
12
01
42
212
401
342
0 32 3 0 -8 8
Sum of red terms = 0 + 32 + 3 = 35
Sum of blue terms = 0 – 8 + 8 = 0
Determinant of matrix A= det(A) = 35 – 0 = 35
The Determinant for a 3x3 Matrix – An Example
Evaluate determinant A by a cofactor along the third column
det(A)=a13C13 +a23C23+a33C33
det(A)=
1 5
1 0
3 -1
-3
2
2
det(A)= -3(-1-0)+2(-1)5(-1-15)+2(0-5)=25
det(A)= 1 0
3 -1
-3(-1)4 1 5
3 -1
+2 (-1)5 1 5
1 0
+2 (-1)6
The Determinant for a 3x3 Matrix – An Example
1
3
-1
0
0
4
5
1
2
0
2
1
-3
1
-2
3
A=
det(A)=(1)
4 0 1 5 2 -2 1 1 3
- (0)
3 0 1 -1 2 -2 0 1 3
+ 2
3 4 1 -1 5 -2 0 1 3
- (-3)
3 4 0 -1 5 2 0 1 1
= (1)(35)-0+(2)(62)-(-3)(13)=198
det(A) = a11C11 +a12C12 + a13C13 +a14C14
The Determinant for a 4x4 Matrix – An Example
Some Properties of determinants
Determinants have several mathematical properties useful in matrix manipulations: |A|=|A'|
If each element of a row (or column) of A is 0, then |A|= 0
If every value in a row is multiplied by k, then |A| = k|A|
If two rows (or columns) are interchanged the sign, but not value, of |A| changes
If two rows (or columns) of A are identical, |A| = 0
Some Properties of Determinants
|A| remains unchanged if each element of a row is multiplied by a constant and added to any other row
If A is nonsingular, then |A|=1/|A-1|, i.e., |A||A-1|=1
|AB|= |A||B| (i.e., the determinant of a product = product of the determinants)
For any scalar c, |cA| = ck|A| where k is the order of A
Determinant of a diagonal matrix is simply the product of the diagonal elements
Why are Determinants Important?
Consider the small system of equations:
a11x1 + a12x2 = b1
a21x1 + a22x2 = b2
Which can be represented by:
Ax = b
where
11 12 1 1
21 22 2 2
a a x b= , = , and =
a a x bA x b
Why are Determinants Important?
If we were to solve this system of equations simultaneously for x2 we would have:
a21(a11x1 + a12x2 = b1)
-a11(a21x1 + a22x2 = b2)
Which yields (through cancellation & rearranging):
a21a11x1 + a21a12x2 - a11a21x1 - a11a22x2 =
a21b1 - a11b2
Why are Determinants Important?
or (a11a2 - a21a12)x2 = a11b2 - a21b1
which implies 11 2 21 1
2
11 22 12 21
a b a bx =
a a - a a
Notice that the denominator is:
11 22 12 21= a a - a aA
Thus iff |A|= 0 there is either i) no unique solution or ii) no existing solution
to the system of equations Ax = b!
Why are Determinants Important?
This result holds true:
if we solve the system for x1 as well; or
for a square matrix A of any order.
Thus we can use determinants in conjunction with the A matrix (coefficient matrix in a system of simultaneous equations) to see if the system has a unique solution.
Show that the determinant of any orthogonal matrix is either +1 or –1.
For any orthogonal matrix, AAT = I.
Since |AAT| = |A||AT | = 1 and |AT| = |A|, so |A|2 = 1 or |A| = 1.
The Determinant of Orthogonal Matrix