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• Angel Chavez

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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.  a11 a12 a13  D  a21 a22 a23   a31 a32 a33  where: aij = the element of the ith row and jth column

Sign of Operators    D     

                      

Properties of Determinants: 1.

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

0 1 4    D  0 2 5   0 0 3 6

2.

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

1 4 7  2 5 8  D  2 5 8    1 4 7  3 6 9  3 6 9 

Properties of Determinants: 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.

 2 4 4  2 1 4 D  4 12 5  4 3 5  4 5 16 6 5 4 6 common factor =4

Properties of Determinants: 4. If two rows or columns are identical then D = 0.

1 1 4    D  2 2 5  0 3 3 6 identical

Properties of Determinants: 5. If two rows or columns are proportional then D is equivalent to 0.

1 2 7  D  2 4 8   0 3 6 9  proportional

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

1 4 7  1 2 3     D   2 5 8   4 5 6 3 6 9  7 8 9

Properties of Determinants: 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

 a11 b11 a12 a  21 b21 a22 a31 b31 a32 D1

a13   a11 a12 a13  b11 a12 a13       a23   a21 a22 a23   b21 a22 a23  a33  a31 a32 a33  b31 a32 a33   D2  D3

Properties of Determinants: 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.

 a11 a12  A  a21 a22 

b11 b12  B  b21 b22 

 (a11b11  a12b21 ) (a11b12  a12b22 )  A* B    (a21b11  a22b21 ) (a21b12  a22b22 )

Properties of Determinants: 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)

Properties of Determinants: Ex. Expansion by Row D = a11 a12 a13 a21 a22 a23 a31 a32 a33 =(+) a11 a22 a23 a32 a33

+ (-) a12 a21 a23 + (+) a13 a21 a31 a33 a31

a22 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 3 1 2 1

EVALUATION OF DETERMINANTS 1. Conventional Method used for 2nd degree determinants and commonly denoted as cross product method.

Ex. D =

2 1 8 5 D = [(2x5)-(8x1)] D = ( 10 - 8 ) = 2

EVALUATION OF DETERMINANTS 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 / .

EVALUATION OF DETERMINANTS Ex.

D

=

2 5 1

5 0 -3

4 1 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

EVALUATION OF DETERMINANTS 3. Expansion by Minor Cofactor Method used for 3rd degree and higher degree order of determinants. a. Expansion by Row n

D   Aik Dik k 1

EVALUATION OF DETERMINANTS Find the determinants of the given matrix using expansion by row. D

=

1 4 2

4 4 5

3 5 4

EVALUATION OF DETERMINANTS b. Expansion by Column n

D   Akj Dkj k 1

EVALUATION OF DETERMINANTS Find the determinants of the given matrix using expansion by column. D

=

1 4 2

4 4 5

3 5 4

EVALUATION OF DETERMINANTS • Chio’s Method – Another method in evaluating the determinant of an (m x m) order matrix where a11 is not equal to zero.

EVALUATION OF DETERMINANTS • Chio’s Method (3x3) a11 A  a 21 a31

a12 a 22 a32

a13 a 23 a33

where: m is the size of the square matrix 33

a11

a12

a 21 1 det . A  (a11 ) ( m  2 ) a11 a31

a 22 a12 a32

a11 a 21 a11 a31

a13 a 23 a13 a33

33

EVALUATION OF DETERMINANTS • Chio’s Method (4x4) A

a11

a12

a13

a14

a 21

a 22

a 23

a 24

a31

a32

a33

a34

a 41

a 42

a 43

a 44

where: m is the size of the square matrix 44

a11

a12

a11

a13

a11

a14

a 21 a11

a 22 a12

a 21 a11

a 23 a13

a 21 a11

a 24 a14

a32 a12

a31 a11

a33 a13

a31 a11

a34 a14

a 42

a 41

a 43

a 41

a 44

1 det . A  (a11 ) ( m  2 ) a31 a11 a 41

44

EVALUATION OF DETERMINANTS Find the determinants of the given matrix using Chio’s method. D

=

1 4 2

4 4 5

3 5 4

Techniques in Altering the Elements of the Determinants. •

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

a. Alteration by zero The 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.

Techniques in Altering the Elements of the Determinants. D

=

A D G

B E H

C F I

D

=

A 0 0

B E 0

C F 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 2

4 4 5

3 5 4

Techniques in Altering the Elements of the Determinants. b.

Pivotal Element Method Steps: 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 2

4 4 5

3 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 3x3 A = /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

Special Type of Matrices 2.

Column Vector Matrix A matrix which contains only one column and several rows. Ex. C=

1 2 3 . . . n

m x1

Special Type of Matrices 3.

Square Matrix It is a matrix whose elements have equal number of rows and columns. Ex. A

=

1 4 2

4 4 5

3 5 4

Special Type of Matrices 4. Null or Zero Matrix It is a square matrix whose elements are all zeros. Ex. A =

0 0

0 0

Special Type of Matrices 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

Special Type of Matrices 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

Special Type of Matrices 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 2

6 2 3

Special Type of Matrices 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 -2

-6 2 3

Special Type of Matrices 9. Singular Matrix It is a square matrix whose determinant value is equivalent to 0. Ex. A =

1 2

4 8

Special Type of Matrices 10. Non-singular Matrix It is a square matrix whose determinant value is not equivalent to 0. Ex. A

=

1 4 2

4 4 5

3 5 4

determinant value /A/ = 3

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 of the two matrices are equal.

Amxn + Bmxn = Cmxn

size

MATRIX OPERATION Example:

Evaluate A & B A= 1 2 3 3 2 1 2x3 B= 1 2 4 1 0 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 + B b. C = B – A A=

1 2 3

4 5 6

7 8 9 3x3

B=

2 1 0

5 6 3

1 4 7 3x3

MATRIX OPERATION 2. Multiplication a. By scalar Example: A x 5 where A = 1 3 4

Ax5= 5 15 20

0 -4 5

1 3 2

0 -20 20

5 15 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 a21 a31

a12 a22 a32

a13 a23 a33

3x3

b11 b21 b31

b12 b22 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 c21 c31

c12 c22 c32

3x2

MATRIX OPERATION 3. Division Inverse of a matrix B=1 A

= A-1

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 2

4 4 5

3 5 4

AT = 1 4 3

4 4 5

2 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 5 4

(-)4 5 2 4

(+)4 4 2 5

(-)4 3 5 4

(+)1 3 2 4

(-)1 4 2 5

(+)4 3 4 5

(-)1 3 4 5

(+)1 4 4 4

MATRIX OPERATION Ex. Solve for the inverse matrix. A=

1 4 2

4 4 5

3 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 A2 A3

B1 B2 B3

C1 C2 C3

: : :

D1 D2 D3

A1 0 0

B1 B2’ 0

C1 C2’ C3’

: : :

D1 D2’ D3’

X1

X2

X3

K

  

1 2 3

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 A2 A3

B1 B2 B3

C1 C2 C3

: : :

D1 D2 D3

1 0 0

0 1 0

0 0 1

: : :

X1 X2 X3

Cramer’s Rule • Use Gauss-Jordan Reduction and GaussJordan 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 x  x M     y  y x  x  0 M         y  y  0 x  x  0 M     I        y  y  0  x  0 M  I        y  0

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…

Eigenvalues and Eigenvectors  x  0  M  I        y  0  a b  M   c d    a b   0   M  I      c d 0      b  a   M  I     c d     a    c 

b   x  0      d     y  0

Write M as a matrix…

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

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

Eigenvalues and Eigenvectors a    c 

b   x  0      d     y  0

 x  a    y   c     x  0  y   0    

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

1

b  0 d     0 

This would mean that the vector was zero.

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

Eigenvalues and Eigenvectors a

b

c

d 

0

This bit is the useful bit to remember

 a    d     bc  0  2   a  d     ad  bc   0 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.

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

 x  x M     y  y

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

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

Eigenvalues and Eigenvectors • Find the eigenvalues and corresponding eigenvectors of matrix A.

A

1 4 2 3