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