Post on 21-Apr-2015
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
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.
common factor =4
4
645
534
412
6165
5124
442
D
Properties of Determinants:
4. If two rows or columns are identical then D = 0.
identical
0
633
522
411
D
Properties of Determinants:
5. If two rows or columns are proportional then D is equivalent to 0.
proportional
0
963
842
721
D
Properties of Determinants:
6. If the corresponding rows and columns of a determinant are interchanged, its value is unchanged.
987
654
321
963
852
741
D
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
321
333231
232221
131211
333231
232221
131211
33323131
23222121
13121111
DDD
aab
aab
aab
aaa
aaa
aaa
aaba
aaba
aaba
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.
2221
1211
aa
aaA
2221
1211
bb
bbB
)()(
)()(*
2222122121221121
2212121121121111
babababa
babababaBA
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 + (-) 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
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 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
EVALUATION OF DETERMINANTS
3. Expansion by Minor Cofactor Method
used for 3rd degree and higher degree order of determinants.
a. Expansion by Row
n
kikikDAD
1
EVALUATION OF DETERMINANTS
Find the determinants of the given matrix using expansion by row.
D = 1 4 3
4 4 5
2 5 4
EVALUATION OF DETERMINANTS
b. Expansion by Column
n
kkjkjDAD
1
EVALUATION OF DETERMINANTS
Find the determinants of the given matrix using expansion by column.
D = 1 4 3
4 4 5
2 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)
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
EVALUATION OF DETERMINANTS
• 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
EVALUATION OF DETERMINANTS
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.
Techniques in Altering the Elements 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
Techniques in Altering the Elements of the Determinants.
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
Special Type of Matrices
2. Column Vector MatrixA matrix which contains only one column and several rows.
Ex.C = 1
23...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 3
4 4 5
2 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 26 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 26 -2 3
Special Type of Matrices
9. Singular Matrix
It is a square matrix whose determinant value is equivalent to 0.
Ex.
A = 1 4
2 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 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…
Eigenvalues and Eigenvectors
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…
Eigenvalues and Eigenvectors
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.
Eigenvalues and Eigenvectors
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
Eigenvalues and Eigenvectors
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!
Eigenvalues and Eigenvectors
• Find the eigenvalues and corresponding eigenvectors of matrix A.
32
41A