Matrix and Determinants(2012)

Matrix and Determinants

Determinants

• a square array of numbers enclosed by two bars and is subjected to mathematical operation. The elements of which have corresponding numbers of rows to that of the columns.

where:

aij = the element of the ith row and jth column

333231

232221

131211

aaa

aaa

aaa

D

Sign of Operators

D

Properties of Determinants:

1. If the value of a single row or column are all 0 then D = 0.

2. If two rows or columns are interchanged, the sign of the determinant is changed

0

630

520

410

D

963

741

852

963

852

741

D


3. If each element of a row or column of a determinant can be multiplied by a common factor then the determinant is multiplied by that number.

common factor =4

4

645

534

412

6165

5124

442

D


4. If two rows or columns are identical then D = 0.

identical

0

633

522

411

D


5. If two rows or columns are proportional then D is equivalent to 0.

proportional

0

963

842

721

D


6. If the corresponding rows and columns of a determinant are interchanged, its value is unchanged.

987

654

321

963

852

741

D


7. If three determinants D1, D2 and D3 have corresponding equal elements except for a single row or column in which the elements at D1 are the sum of the corresponding elements of D2 and D3 then D1 = D2 + D3

321

333231

232221

131211

333231

232221

131211

33323131

23222121

13121111

DDD

aab

aab

aab

aaa

aaa

aaa

aaba

aaba

aaba


8. The product of two determinants of the same order is also a determinant of the same order whose element in the ith row and jth column is the sum of the products of the ith row of the first determinant and the jth column of the second determinant.

2221

1211

aa

aaA

2221

1211

bb

bbB

)()(

)()(*

2222122121221121

2212121121121111

babababa

babababaBA


9. The value of a determinant is the algebraic sum of the products obtained by multiplying each element of a column or row by its co-factor or signed minor.

(Expansion of Determinants by Minor)


Ex. Expansion by Row

D = a11 a12 a13

a21 a22 a23

a31 a32 a33

=(+) a11 a22 a23 + (-) a12 a21 a23 + (+) a13 a21 a22

a32 a33 a31 a33 a31 a32

Minor and Cofactors

• The Minor of the element Aij in the ith row and jth column in any determinant order formed from the element remained after isolating the ith row and jth column.

M13 = a21 a22

a31 a32

M23 = a11 a12

a31 a32

where :

Mij = the minor of Aij

Minor and Cofactors

• The cofactor of the element Aij in any determinant of order n is that signed minor determined by,

Dij = ( -1 )i + j ( Mij )

D13 = ( -1 )1+3 ( M13 )

= ( +1 ) a21 a22

a31 a32

D23 = ( -1 )2+3 ( M23 )

= ( -1 ) a11 a12

a31 a32

where :Dij = the cofactor of Aij

Example

Find the cofactor using the minor of the given matrix

D = 1 2 3

-2 3 1

3 2 1

EVALUATION OF DETERMINANTS

1. Conventional Methodused for 2nd degree determinants and commonly denoted as cross product method.

Ex. D = 2 1

8 5D = [ ( 2 x 5 ) - ( 8 x 1 ) ]D = ( 10 - 8 ) = 2


2. Diagonal Method

used for 3rd degree determinants commonly described as the sum of products of the diagonal leaning \ minus the sum of products of the diagonal leaning / .


Ex.D = 2 5 4

5 0 11 -3 3

D = [( 2 x 0 x 3 ) + ( 5 x 1 x 1 ) + ( 4 x 5 x -3 )] - [( 1 x 0 x 4 ) + ( -3 x 1 x 2 ) + ( 3 x 5 x 5 )]D = ( 0 + 5 + -60 ) - ( 0 + -6 + 75 )D = - 55 - 69 = - 124


3. Expansion by Minor Cofactor Method

used for 3rd degree and higher degree order of determinants.

a. Expansion by Row

n

kikikDAD

1


Find the determinants of the given matrix using expansion by row.

D = 1 4 3

4 4 5

2 5 4


b. Expansion by Column

n

kkjkjDAD

1


Find the determinants of the given matrix using expansion by column.

D = 1 4 3

4 4 5

2 5 4


• Chio’s Method

– Another method in evaluating the determinant of an (m x m) order matrix where a11 is not equal to zero.


• Chio’s Method (3x3)

333331

1311

3231

1211

2321

1311

2221

1211

)2(11

33333231

232221

131211

)(

1.det

aa

aa

aa

aaaa

aa

aa

aa

aA

aaa

aaa

aaa

A

m

where: m is the size of the square matrix


• Chio’s Method (4x4)

where: m is the size of the square matrix

444441

1411

4341

1311

4241

1211

3431

1411

3331

1311

3231

1211

2421

1411

2321

1311

2221

1211

)2(11

4444434241

34333231

24232221

14131211

)(

1.det

aa

aa

aa

aa

aa

aaaa

aa

aa

aa

aa

aaaa

aa

aa

aa

aa

aa

aA

aaaa

aaaa

aaaa

aaaa

A

m


Find the determinants of the given matrix using Chio’s method.

D = 1 4 3

4 4 5

2 5 4

Techniques in Altering the Elements of the Determinants.

• used for 3rd degree and higher degree order of determinants.

a. Alteration by zeroThe element of any row (or column) may be multiplied by a constant and the result added to the corresponding element of any other row (or column) without changing the value of the determinants.


D = A B C D E F G H I

D = A B C 0 E F 0 0 I

D = ( A x E x I )

Techniques in Altering the Elements of the Determinants.Find the determinants of the given matrix using alteration by zero.

D = 1 4 3

4 4 5

2 5 4


b. Pivotal Element MethodSteps:1. Select a pivot element except zero.

2. Draw cancellation lines along the row and column of the pivotal element.

3. Replace the remaining element by subtracting from the original element, the product of the elements intersecting the cancellation lines and perpendicular lines dividing it by the pivot element.

4. Multiply the resulting determinant by the pivot element along with its corresponding sign of cofactor.

Techniques in Altering the Elements of the Determinants.Find the determinants of the given matrix using pivot element method.

D = 1 4 3

4 4 5

2 5 4

Matrix

• It is a rectangular array of numbers or functions enclosed in a pair of brackets and subject to certain rules of operation.

A = a11 a12 a13

a21 a22 a23

a31 a32 a33 3x3A = /aij/mxn

where: aij = element of matrix A mxn = size of order of matrix

m = number of row matrix n = number of column matrix

Note: m & n may or may not be equal

Special Type of Matrices

1. Row Vector Matrix

A matrix which contains only one row and several columns.

Ex.

B = [ 1 2 3 4 ….. n ] 1 x n


2. Column Vector MatrixA matrix which contains only one column and several rows.

Ex.C = 1

23...n m x1


3. Square Matrix

It is a matrix whose elements have equal number of rows and columns.

Ex.

A = 1 4 3

4 4 5

2 5 4


4. Null or Zero Matrix

It is a square matrix whose elements are all zeros.

Ex.

A = 0 0

0 0


5. Diagonal Matrix

It is a square matrix wherein the values lie in the main diagonal and the rest are all zeros.

Ex.

A = 3 0 0

0 4 0

0 0 3


6. Unity or Identity Matrix

It is a square matrix whose elements in the main diagonal are all 1’s and the rest are all zeros.

Ex.

A = 1 0 0

0 1 0

0 0 1


7. Symmetric Matrix

It is a square matrix whose element Aij is equal to the element Aji or the elements of the rows corresponds to that of the column.

Ex.

A = 1 -5 6-5 7 26 2 3


8. Skew Matrix

It is a square matrix whose element Aij is equal to the negative of the element Aji.

Ex.

A = 1 5 -6-5 7 26 -2 3


9. Singular Matrix

It is a square matrix whose determinant value is equivalent to 0.

Ex.

A = 1 4

2 8


10. Non-singular Matrix

It is a square matrix whose determinant value is not equivalent to 0.

Ex.

A = 1 4 3 determinant value

4 4 5 /A/ = 3

2 5 4

MATRIX LAWS

1. A + B = B + A (Matrix addition is commutative)

2. A + 0 = 0 + A = A (0 is the zero for matrix addition)

3. A + (-A) = (-A) + A = 0 (-A is the negative of A)4. (A + B) + C = A + (B + C)(Matrix addition is associative)5. (sA)B = A (sB) = s(AB) (Scalars can be moved

through products)6. For Amxn, ImA = Ain = A (I is the identity for matrix

multiplication)7. (A + B)C = AC + BC (Right distributive law)8. A(B + C) = AB + AC (Left distributive law)9. A(BC) = (AB) C (Matrix multiplication is

associative)

MATRIX OPERATION

• In matrix operation, only addition, subtraction, and multiplication are defined. Division is done by a different technique.

1. Addition / Subtraction

-two matrices may be conformable to addition/subtraction, if and only if the size of the two matrices are equal.

Amxn + Bmxn = Cmxn

MATRIX OPERATION

Example:Evaluate A & B

A = 1 2 33 2 1 2x3

B = 1 24 10 1 3x2

A + B = not possible because they are not of the same order.

MATRIX OPERATION

Example:

Find the sum and difference of the two matrices based on the following conditions:a. C = A + Bb. C = B – A

A = 1 4 7 B = 2 5 12 5 8 1 6 43 6 9 3x3 0 3 7 3x3

MATRIX OPERATION

2. Multiplication

a. By scalarExample:

A x 5 where A = 1 0 1

3 -4 3

4 5 2

A x 5 = 5 0 5

15 -20 15

20 20 10

MATRIX OPERATION

b. By another matrix

-two matrices can be multiplied if the number of column (left hand) of the first matrix is equal to the number of row (right hand) of the second matrix.

MATRIX OPERATION

A * B = a11 a12 a13 b11 b12

a21 a22 a23 b21 b22

a31 a32 a33 3x3 b31 b32 3x2

MATRIX OPERATION

A * B = (a11b11+a12b21+a13b31) (a11b12+a12b22+a13b32)

C 11 C 12

(a21b11+a22b21+a23b31) (a21b12+a22b22+a23b32)

C 21 C 22

(a31b11+a32b21+a33b31) (a31b12+a32b22+a33b32)

C 31 C 32

MATRIX OPERATION

A * B = C = c11 c12

c21 c22

c31 c32 3x2

MATRIX OPERATION

3. Division

Inverse of a matrix

B = 1 = A-1

A

where: A-1 inverse of matrix A

MATRIX OPERATION

a. Inverse of a matrix

A-1 = 1 adj AT

/A/

where: /A/ = determinant value of a matrix

AT = transpose of a matrix

adj = adjoint of a matrix

Note: Division operation is not conformable to matrices of unequal rows & columns. Hence, this operation is restricted to square matrices only.

MATRIX OPERATION

Transpose of a matrix ( AT )

-to determine the transpose of a matrix, interchange the corresponding rows and columns of the given determinant.

Example:

A = 1 4 3 AT = 1 4 2

4 4 5 4 4 5

2 5 4 3 5 4

MATRIX OPERATION

Adjoint of a Matrix ( adj. )

-to obtain the adjoint of any matrix, replace each element by its corresponding co-factor.

Example:

A = 1 4 3

4 4 5

2 5 4

MATRIX OPERATION

adj A = (+)4 5 (-)4 5 (+)4 4

5 4 2 4 2 5

(-)4 3 (+)1 3 (-)1 4

5 4 2 4 2 5

(+)4 3 (-)1 3 (+)1 4

4 5 4 5 4 4

MATRIX OPERATION

Ex. Solve for the inverse matrix.

A = 1 4 3

4 4 5

2 5 4

Solutions to Linear Equations

• Cramers Rule

• Gauss-Jordan Methods

Cramer’s Rule

• It is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution.

• The solution is expressed in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations.

Cramer’s Rule

• Use Cramer’s Rule to solve the system:

4x - y + z = -5

2x + 2y + 3z = 10

5x – 2y + 6z = 1

GAUSS-JORDAN METHODS

• Gauss-Jordan Elimination

• Gauss-Jordan Reduction

Gauss-Jordan Reduction

• To obtain the values of the unknown variables in a given linear equation: plot the constants along with the coefficients of the unknown variables and then apply alteration by zero producing simplified equations to solve for the unknown variables.

Gauss-Jordan Reduction

A1 X1 + B1 X2 + C1 X3 = D1

A2 X1 + B2 X2 + C2 X3 = D2

A3 X1 + B3 X2 + C3 X3 = D3

A1 B1 C1 : D1

A2 B2 C2 : D2

A3 B3 C3 : D3

A1 B1 C1 : D1 10 B2’ C2’ : D2’ 20 0 C3’ : D3’ 3

X1 X2 X3 K

Gauss-Jordan Elimination

• To obtain the values of the unknown variables in a given linear equation: plot the constants along with the coefficients of the unknown variables and then apply row-by-row transformation to change the given matrix to a unity matrix thus, altering the value of the constants yielding the values of the unknown variables.

Gauss-Jordan Elimination

A1 X1 + B1 X2 + C1 X3 = D1

A2 X1 + B2 X2 + C2 X3 = D2

A3 X1 + B3 X2 + C3 X3 = D3

A1 B1 C1 : D1

A2 B2 C2 : D2

A3 B3 C3 : D3

1 0 0 : X1

0 1 0 : X2

0 0 1 : X3

Cramer’s Rule

• Use Gauss-Jordan Reduction and Gauss-Jordan Elimination to solve the given system:

4x - y + z = -5

2x + 2y + 3z = 10

5x – 2y + 6z = 1

Eigenvalues and Eigenvectors of a Matrix

Eigenvalues and Eigenvectors

0

0

0

0

0

0

x xM

y y

x xM

y y

x xM I

y y

xM I

y

M is a matrix and λ is a scalar constant

Rearranging…

In order to factorise scalar λ must turn into a matrix by multiplying it by the identity matrix.

now it can be factorise…


0

0

0

0

0

0

xM I

y

a bM

c d

a bM I

c d

a bM I

c d

a b x

c d y

If the determinant of (M- λI) was non-zero, it could be inverse and multiplied by the RHS.

Write (M- λI) in the following way and then simplify the equation.

Write M as a matrix…


1

0

0

0

0

0

0

a b x

c d y

x a b

y c d

x

y

If the determinant of (M- λI) was non-zero, it could be inverse and multiply it by the RHS.

This would mean that the vector was zero.

This means that the determinant of (M- λI) must be zero so is singular.


2

0

0

0

a b

c d

a d bc

a d ad bc

This is called the characteristic equations and will allow to find the eigenvalues (characteristic values)

The Eigenvalues represent the scale factor of the distance our position vector (which we still need to find) moves in relation to its original position. There may be 1, 2 or no real eigenvalues in a 2x2 matrix.

This bit is the useful bit to remember


x xM

y y

Once the Eigenvalues have found, substitute these back to find their corresponding Eigenvectors.

Eigenvectors represent Invariant Lines. These are the lines of points that map onto themselves after a transformation.

This represents the Eigenvector. It is not unique as any multiple of it would still be an Eigenvector!


• Find the eigenvalues and corresponding eigenvectors of matrix A.

32

41A

Matrix and Determinants(2012)

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