Matrices and Determinants were discovered and developed in the eighteenth and nineteenth centuries. Initially, their development dealt with transformation of geometric objects and solution of systems of linear equations. Historically, the early emphasis was on the determinant, not the matrix. In modern treatments of linear algebra, matrices are considered first. We will not speculate much on this issue.
Matrices provide a theoretically and practically useful way of approaching many types of problems including:
Solution of Systems of Linear Equations,
Equilibrium of Rigid Bodies (in physics),
Graph Theory,
Theory of Games,
Leontief Economics Model,
Forest Management,
Computer Graphics, and Computed Tomography,
Genetics,
Cryptography,
Electrical Networks,
Fractals.
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Sunday, August 1, 2010
Matrix
Matrix
A matrix is an ordered set of numbers listed rectangular form.
Example. Let A denote the matrix
[2 5 7 8]
[5 6 8 9]
[3 9 0 1]
This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.
We denote the element on the second row and fourth column with a2,4.
Square matrix
If a matrix A has n rows and n columns then we say it's a square matrix.
In a square matrix the elements ai,i , with i = 1,2,3,... , are called diagonal elements.
Remark. There is no difference between a 1 x 1 matrix and an ordenary number.
Diagonal matrix
A diagonal matrix is a square matrix with all de non-diagonal elements 0.
The diagonal matrix is completely denoted by the diagonal elements.
Example.
[7 0 0]
[0 5 0]
[0 0 6]
The matrix is denoted by diag(7 , 5 , 6)
Row matrix
A matrix with one row is called a row matrix
Column matrix
A matrix with one column is called a column matrix
Matrices of the same kind
Matrix A and B are of the same kind if and only if
A has as many rows as B and A has as many columns as B
The tranpose of a matrix
The n x m matrix A' is the transpose of the m x n matrix A if and only if
The ith row of A = the ith column of A' for (i = 1,2,3,..n)
So ai,j = aj,i'
The transpose of A is denoted T(A) or AT
0-matrix
When all the elements of a matrix A are 0, we call A a 0-matrix.
We write shortly 0 for a 0-matrix.
An identity matrix I
An identity matrix I is a diagonal matrix with all diagonal element = 1.
A scalar matrix S
A scalar matrix S is a diagonal matrix with all diagonal elements alike.
a1,1 = ai,i for (i = 1,2,3,..n)
The opposite matrix of a matrix
If we change the sign of all the elements of a matrix A, we have the opposite matrix -A.
If A' is the opposite of A then ai,j' = -ai,j, for all i and j.
A symmetric matrix
A square matrix is called symmetric if it is equal to its transpose.
Then ai,j = aj,i , for all i and j.
A skew-symmetric matrix
A square matrix is called skew-symmetric if it is equal to the opposite of its transpose.
Then ai,j = -aj,i , for all i and j.
The sum of matrices of the same kind
Sum of matrices
To add two matrices of the same kind, we simply add the corresponding elements.
Sum properties
Consider the set S of all n x m matrices (n and m fixed) and A and B are in S.
From the properties of real numbers it's immediate that
A + B is in S
the addition of matrices is associative in S
A + 0 = A = 0 + A
with each A corresponds an opposite matrix -A
A + B = B + A
Scalar multiplication
Definition
To multiply a matrix with a real number, we multiply each element with this number.
Properties
Consider the set S of all n x m matrices (n and m fixed). A and B are in S; r and s are real numbers.
It is not difficult to see that:
r(A+B) = rA+rB
(r+s)A = rA+sA
(rs)A = r(sA)
(A + B)T = AT + BT
(rA)T = r. AT
Sums in math
Because in the following, there is an intensive use of the properties of sums, the reader who is not familiar with these properties must read first Sums in math .
Remark. In this html document, for convenience, we'll write the word sum instead of the sigma sign.
Multiplication of a row matrix by a column matrix
This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).
To multiply the row by the column, one multiplies corresponding elements, then adds the results.
Example.
[1]
[2 1 3]. [2] = [19]
[5]
Multiplication of two matrices A.B
This product is defined only if A is a (l x m) matrix and B is a (m x n) matrix.
So the number of columns of A has to be equal to the number of rows of B.
The product C = A.B then is a (l x n) matrix.
The element of the ith row and the jth column of the product is found by multiplying the ith row of A by the jth column of B.
ci,j = sumk (ai,k.bk,j)
Example.
[1 2][1 3] = [5 7]
[2 1][2 2] [4 8]
[1 3][1 2] = [7 5]
[2 2][2 1] [6 6]
[1 1][2 2] = [0 0]
[1 1][-2 -2] [0 0]
From these examples we see that the product is not commutative and that there are zero divisors.
Properties of multiplication of matrices
Associativity
If the multiplication is defined then A(B.C) = (A.B)C holds for all matrices A,B and C.
Proof:
We'll show that an element of A(B.C) is equal to the corresponding element of (A.B)C
First we calculate the element of the ith row and jth column of A(B.C)
Let D denote B.C, then
dk,j = sump bk,p.cp,j (1)
Let E denote A.D then
ei,j = sumk ai,k.dk,j (2)
(1) in (2) gives
ei,j = sumk ai,k.(sump bk,p.cp,j)
<=> ei,j = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of A(B.C) is
sumk,p ai,k.bk,p.cp,j (3)
Now we calculate the element of the ith row and jth column of (A.B)C
Let D' denote A.B, then
di,p' = sumk ai,k.bk,p (4)
Let E' denote D'C then
ei,j' = sump di,p'.cp,j (5)
(4) in (5) gives
ei,j' = sump (sumk ai,k.bk,p).cp,j
<=> ei,j' = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of (A.B)C is
sumk,p ai,k.bk,p.cp,j (6)
From (3) and (6) => A(B.C) = (A.B)C
Distributivity
If the multiplication is defined then A(B+C) = A.B+A.C and (A+B).C = A.C+B.C holds for all matrices A,B and C. This theorem can be proved in the same way as above.
Theorem 1
For each A, there is always an identity matrix E and an identity matrix E' so that A.E = A and E'.A = A If A is a square matrix, E = E'.
Theorem 2
(A.B)T = BT .AT
This theorem can be proved in the same way as above.
Theorem 3
If the multiplication is defined then for each A
A.0 = 0 = 0.A
Theorem 4
r and s are real numbers and A , B matrices. If the multiplication is defined then (rA)(sB) = (rs)(AB) This theorem can be proved in the same way as above.
Theorem 5
if D = diag(a,b,c) then D.D = ( a2 , b2 , c2)
D.D.D = ( a3 , b3 , c3)
.....
This property can be generalised for D = diag(a,b,c,d,e,...,l).trix is an ordered set of numbers listed rectangular form.
Example. Let A denote the matrix
[2 5 7 8]
[5 6 8 9]
[3 9 0 1]
This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.
We denote the element on the second row and fourth column with a2,4.
Square matrix
If a matrix A has n rows and n columns then we say it's a square matrix.
In a square matrix the elements ai,i , with i = 1,2,3,... , are called diagonal elements.
Remark. There is no difference between a 1 x 1 matrix and an ordenary number.
Diagonal matrix
A diagonal matrix is a square matrix with all de non-diagonal elements 0.
The diagonal matrix is completely denoted by the diagonal elements.
Example.
[7 0 0]
[0 5 0]
[0 0 6]
The matrix is denoted by diag(7 , 5 , 6)
Row matrix
A matrix with one row is called a row matrix
Column matrix
A matrix with one column is called a column matrix
Matrices of the same kind
Matrix A and B are of the same kind if and only if
A has as many rows as B and A has as many columns as B
The tranpose of a matrix
The n x m matrix A' is the transpose of the m x n matrix A if and only if
The ith row of A = the ith column of A' for (i = 1,2,3,..n)
So ai,j = aj,i'
The transpose of A is denoted T(A) or AT
0-matrix
When all the elements of a matrix A are 0, we call A a 0-matrix.
We write shortly 0 for a 0-matrix.
An identity matrix I
An identity matrix I is a diagonal matrix with all diagonal element = 1.
A scalar matrix S
A scalar matrix S is a diagonal matrix with all diagonal elements alike.
a1,1 = ai,i for (i = 1,2,3,..n)
The opposite matrix of a matrix
If we change the sign of all the elements of a matrix A, we have the opposite matrix -A.
If A' is the opposite of A then ai,j' = -ai,j, for all i and j.
A symmetric matrix
A square matrix is called symmetric if it is equal to its transpose.
Then ai,j = aj,i , for all i and j.
A skew-symmetric matrix
A square matrix is called skew-symmetric if it is equal to the opposite of its transpose.
Then ai,j = -aj,i , for all i and j.
The sum of matrices of the same kind
Sum of matrices
To add two matrices of the same kind, we simply add the corresponding elements.
Sum properties
Consider the set S of all n x m matrices (n and m fixed) and A and B are in S.
From the properties of real numbers it's immediate that
A + B is in S
the addition of matrices is associative in S
A + 0 = A = 0 + A
with each A corresponds an opposite matrix -A
A + B = B + A
Scalar multiplication
Definition
To multiply a matrix with a real number, we multiply each element with this number.
Properties
Consider the set S of all n x m matrices (n and m fixed). A and B are in S; r and s are real numbers.
It is not difficult to see that:
r(A+B) = rA+rB
(r+s)A = rA+sA
(rs)A = r(sA)
(A + B)T = AT + BT
(rA)T = r. AT
Sums in math
Because in the following, there is an intensive use of the properties of sums, the reader who is not familiar with these properties must read first Sums in math .
Remark. In this html document, for convenience, we'll write the word sum instead of the sigma sign.
Multiplication of a row matrix by a column matrix
This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).
To multiply the row by the column, one multiplies corresponding elements, then adds the results.
Example.
[1]
[2 1 3]. [2] = [19]
[5]
Multiplication of two matrices A.B
This product is defined only if A is a (l x m) matrix and B is a (m x n) matrix.
So the number of columns of A has to be equal to the number of rows of B.
The product C = A.B then is a (l x n) matrix.
The element of the ith row and the jth column of the product is found by multiplying the ith row of A by the jth column of B.
ci,j = sumk (ai,k.bk,j)
Example.
[1 2][1 3] = [5 7]
[2 1][2 2] [4 8]
[1 3][1 2] = [7 5]
[2 2][2 1] [6 6]
[1 1][2 2] = [0 0]
[1 1][-2 -2] [0 0]
From these examples we see that the product is not commutative and that there are zero divisors.
Properties of multiplication of matrices
Associativity
If the multiplication is defined then A(B.C) = (A.B)C holds for all matrices A,B and C.
Proof:
We'll show that an element of A(B.C) is equal to the corresponding element of (A.B)C
First we calculate the element of the ith row and jth column of A(B.C)
Let D denote B.C, then
dk,j = sump bk,p.cp,j (1)
Let E denote A.D then
ei,j = sumk ai,k.dk,j (2)
(1) in (2) gives
ei,j = sumk ai,k.(sump bk,p.cp,j)
<=> ei,j = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of A(B.C) is
sumk,p ai,k.bk,p.cp,j (3)
Now we calculate the element of the ith row and jth column of (A.B)C
Let D' denote A.B, then
di,p' = sumk ai,k.bk,p (4)
Let E' denote D'C then
ei,j' = sump di,p'.cp,j (5)
(4) in (5) gives
ei,j' = sump (sumk ai,k.bk,p).cp,j
<=> ei,j' = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of (A.B)C is
sumk,p ai,k.bk,p.cp,j (6)
From (3) and (6) => A(B.C) = (A.B)C
Distributivity
If the multiplication is defined then A(B+C) = A.B+A.C and (A+B).C = A.C+B.C holds for all matrices A,B and C. This theorem can be proved in the same way as above.
Theorem 1
For each A, there is always an identity matrix E and an identity matrix E' so that A.E = A and E'.A = A If A is a square matrix, E = E'.
Theorem 2
(A.B)T = BT .AT
This theorem can be proved in the same way as above.
Theorem 3
If the multiplication is defined then for each A
A.0 = 0 = 0.A
Theorem 4
r and s are real numbers and A , B matrices. If the multiplication is defined then (rA)(sB) = (rs)(AB) This theorem can be proved in the same way as above.
Theorem 5
if D = diag(a,b,c) then D.D = ( a2 , b2 , c2)
D.D.D = ( a3 , b3 , c3)
.....
This property can be generalised for D = diag(a,b,c,d,e,...,l).
A matrix is an ordered set of numbers listed rectangular form.
Example. Let A denote the matrix
[2 5 7 8]
[5 6 8 9]
[3 9 0 1]
This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.
We denote the element on the second row and fourth column with a2,4.
Square matrix
If a matrix A has n rows and n columns then we say it's a square matrix.
In a square matrix the elements ai,i , with i = 1,2,3,... , are called diagonal elements.
Remark. There is no difference between a 1 x 1 matrix and an ordenary number.
Diagonal matrix
A diagonal matrix is a square matrix with all de non-diagonal elements 0.
The diagonal matrix is completely denoted by the diagonal elements.
Example.
[7 0 0]
[0 5 0]
[0 0 6]
The matrix is denoted by diag(7 , 5 , 6)
Row matrix
A matrix with one row is called a row matrix
Column matrix
A matrix with one column is called a column matrix
Matrices of the same kind
Matrix A and B are of the same kind if and only if
A has as many rows as B and A has as many columns as B
The tranpose of a matrix
The n x m matrix A' is the transpose of the m x n matrix A if and only if
The ith row of A = the ith column of A' for (i = 1,2,3,..n)
So ai,j = aj,i'
The transpose of A is denoted T(A) or AT
0-matrix
When all the elements of a matrix A are 0, we call A a 0-matrix.
We write shortly 0 for a 0-matrix.
An identity matrix I
An identity matrix I is a diagonal matrix with all diagonal element = 1.
A scalar matrix S
A scalar matrix S is a diagonal matrix with all diagonal elements alike.
a1,1 = ai,i for (i = 1,2,3,..n)
The opposite matrix of a matrix
If we change the sign of all the elements of a matrix A, we have the opposite matrix -A.
If A' is the opposite of A then ai,j' = -ai,j, for all i and j.
A symmetric matrix
A square matrix is called symmetric if it is equal to its transpose.
Then ai,j = aj,i , for all i and j.
A skew-symmetric matrix
A square matrix is called skew-symmetric if it is equal to the opposite of its transpose.
Then ai,j = -aj,i , for all i and j.
The sum of matrices of the same kind
Sum of matrices
To add two matrices of the same kind, we simply add the corresponding elements.
Sum properties
Consider the set S of all n x m matrices (n and m fixed) and A and B are in S.
From the properties of real numbers it's immediate that
A + B is in S
the addition of matrices is associative in S
A + 0 = A = 0 + A
with each A corresponds an opposite matrix -A
A + B = B + A
Scalar multiplication
Definition
To multiply a matrix with a real number, we multiply each element with this number.
Properties
Consider the set S of all n x m matrices (n and m fixed). A and B are in S; r and s are real numbers.
It is not difficult to see that:
r(A+B) = rA+rB
(r+s)A = rA+sA
(rs)A = r(sA)
(A + B)T = AT + BT
(rA)T = r. AT
Sums in math
Because in the following, there is an intensive use of the properties of sums, the reader who is not familiar with these properties must read first Sums in math .
Remark. In this html document, for convenience, we'll write the word sum instead of the sigma sign.
Multiplication of a row matrix by a column matrix
This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).
To multiply the row by the column, one multiplies corresponding elements, then adds the results.
Example.
[1]
[2 1 3]. [2] = [19]
[5]
Multiplication of two matrices A.B
This product is defined only if A is a (l x m) matrix and B is a (m x n) matrix.
So the number of columns of A has to be equal to the number of rows of B.
The product C = A.B then is a (l x n) matrix.
The element of the ith row and the jth column of the product is found by multiplying the ith row of A by the jth column of B.
ci,j = sumk (ai,k.bk,j)
Example.
[1 2][1 3] = [5 7]
[2 1][2 2] [4 8]
[1 3][1 2] = [7 5]
[2 2][2 1] [6 6]
[1 1][2 2] = [0 0]
[1 1][-2 -2] [0 0]
From these examples we see that the product is not commutative and that there are zero divisors.
Properties of multiplication of matrices
Associativity
If the multiplication is defined then A(B.C) = (A.B)C holds for all matrices A,B and C.
Proof:
We'll show that an element of A(B.C) is equal to the corresponding element of (A.B)C
First we calculate the element of the ith row and jth column of A(B.C)
Let D denote B.C, then
dk,j = sump bk,p.cp,j (1)
Let E denote A.D then
ei,j = sumk ai,k.dk,j (2)
(1) in (2) gives
ei,j = sumk ai,k.(sump bk,p.cp,j)
<=> ei,j = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of A(B.C) is
sumk,p ai,k.bk,p.cp,j (3)
Now we calculate the element of the ith row and jth column of (A.B)C
Let D' denote A.B, then
di,p' = sumk ai,k.bk,p (4)
Let E' denote D'C then
ei,j' = sump di,p'.cp,j (5)
(4) in (5) gives
ei,j' = sump (sumk ai,k.bk,p).cp,j
<=> ei,j' = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of (A.B)C is
sumk,p ai,k.bk,p.cp,j (6)
From (3) and (6) => A(B.C) = (A.B)C
Distributivity
If the multiplication is defined then A(B+C) = A.B+A.C and (A+B).C = A.C+B.C holds for all matrices A,B and C. This theorem can be proved in the same way as above.
Theorem 1
For each A, there is always an identity matrix E and an identity matrix E' so that A.E = A and E'.A = A If A is a square matrix, E = E'.
Theorem 2
(A.B)T = BT .AT
This theorem can be proved in the same way as above.
Theorem 3
If the multiplication is defined then for each A
A.0 = 0 = 0.A
Theorem 4
r and s are real numbers and A , B matrices. If the multiplication is defined then (rA)(sB) = (rs)(AB) This theorem can be proved in the same way as above.
Theorem 5
if D = diag(a,b,c) then D.D = ( a2 , b2 , c2)
D.D.D = ( a3 , b3 , c3)
.....
This property can be generalised for D = diag(a,b,c,d,e,...,l).trix is an ordered set of numbers listed rectangular form.
Example. Let A denote the matrix
[2 5 7 8]
[5 6 8 9]
[3 9 0 1]
This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.
We denote the element on the second row and fourth column with a2,4.
Square matrix
If a matrix A has n rows and n columns then we say it's a square matrix.
In a square matrix the elements ai,i , with i = 1,2,3,... , are called diagonal elements.
Remark. There is no difference between a 1 x 1 matrix and an ordenary number.
Diagonal matrix
A diagonal matrix is a square matrix with all de non-diagonal elements 0.
The diagonal matrix is completely denoted by the diagonal elements.
Example.
[7 0 0]
[0 5 0]
[0 0 6]
The matrix is denoted by diag(7 , 5 , 6)
Row matrix
A matrix with one row is called a row matrix
Column matrix
A matrix with one column is called a column matrix
Matrices of the same kind
Matrix A and B are of the same kind if and only if
A has as many rows as B and A has as many columns as B
The tranpose of a matrix
The n x m matrix A' is the transpose of the m x n matrix A if and only if
The ith row of A = the ith column of A' for (i = 1,2,3,..n)
So ai,j = aj,i'
The transpose of A is denoted T(A) or AT
0-matrix
When all the elements of a matrix A are 0, we call A a 0-matrix.
We write shortly 0 for a 0-matrix.
An identity matrix I
An identity matrix I is a diagonal matrix with all diagonal element = 1.
A scalar matrix S
A scalar matrix S is a diagonal matrix with all diagonal elements alike.
a1,1 = ai,i for (i = 1,2,3,..n)
The opposite matrix of a matrix
If we change the sign of all the elements of a matrix A, we have the opposite matrix -A.
If A' is the opposite of A then ai,j' = -ai,j, for all i and j.
A symmetric matrix
A square matrix is called symmetric if it is equal to its transpose.
Then ai,j = aj,i , for all i and j.
A skew-symmetric matrix
A square matrix is called skew-symmetric if it is equal to the opposite of its transpose.
Then ai,j = -aj,i , for all i and j.
The sum of matrices of the same kind
Sum of matrices
To add two matrices of the same kind, we simply add the corresponding elements.
Sum properties
Consider the set S of all n x m matrices (n and m fixed) and A and B are in S.
From the properties of real numbers it's immediate that
A + B is in S
the addition of matrices is associative in S
A + 0 = A = 0 + A
with each A corresponds an opposite matrix -A
A + B = B + A
Scalar multiplication
Definition
To multiply a matrix with a real number, we multiply each element with this number.
Properties
Consider the set S of all n x m matrices (n and m fixed). A and B are in S; r and s are real numbers.
It is not difficult to see that:
r(A+B) = rA+rB
(r+s)A = rA+sA
(rs)A = r(sA)
(A + B)T = AT + BT
(rA)T = r. AT
Sums in math
Because in the following, there is an intensive use of the properties of sums, the reader who is not familiar with these properties must read first Sums in math .
Remark. In this html document, for convenience, we'll write the word sum instead of the sigma sign.
Multiplication of a row matrix by a column matrix
This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).
To multiply the row by the column, one multiplies corresponding elements, then adds the results.
Example.
[1]
[2 1 3]. [2] = [19]
[5]
Multiplication of two matrices A.B
This product is defined only if A is a (l x m) matrix and B is a (m x n) matrix.
So the number of columns of A has to be equal to the number of rows of B.
The product C = A.B then is a (l x n) matrix.
The element of the ith row and the jth column of the product is found by multiplying the ith row of A by the jth column of B.
ci,j = sumk (ai,k.bk,j)
Example.
[1 2][1 3] = [5 7]
[2 1][2 2] [4 8]
[1 3][1 2] = [7 5]
[2 2][2 1] [6 6]
[1 1][2 2] = [0 0]
[1 1][-2 -2] [0 0]
From these examples we see that the product is not commutative and that there are zero divisors.
Properties of multiplication of matrices
Associativity
If the multiplication is defined then A(B.C) = (A.B)C holds for all matrices A,B and C.
Proof:
We'll show that an element of A(B.C) is equal to the corresponding element of (A.B)C
First we calculate the element of the ith row and jth column of A(B.C)
Let D denote B.C, then
dk,j = sump bk,p.cp,j (1)
Let E denote A.D then
ei,j = sumk ai,k.dk,j (2)
(1) in (2) gives
ei,j = sumk ai,k.(sump bk,p.cp,j)
<=> ei,j = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of A(B.C) is
sumk,p ai,k.bk,p.cp,j (3)
Now we calculate the element of the ith row and jth column of (A.B)C
Let D' denote A.B, then
di,p' = sumk ai,k.bk,p (4)
Let E' denote D'C then
ei,j' = sump di,p'.cp,j (5)
(4) in (5) gives
ei,j' = sump (sumk ai,k.bk,p).cp,j
<=> ei,j' = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of (A.B)C is
sumk,p ai,k.bk,p.cp,j (6)
From (3) and (6) => A(B.C) = (A.B)C
Distributivity
If the multiplication is defined then A(B+C) = A.B+A.C and (A+B).C = A.C+B.C holds for all matrices A,B and C. This theorem can be proved in the same way as above.
Theorem 1
For each A, there is always an identity matrix E and an identity matrix E' so that A.E = A and E'.A = A If A is a square matrix, E = E'.
Theorem 2
(A.B)T = BT .AT
This theorem can be proved in the same way as above.
Theorem 3
If the multiplication is defined then for each A
A.0 = 0 = 0.A
Theorem 4
r and s are real numbers and A , B matrices. If the multiplication is defined then (rA)(sB) = (rs)(AB) This theorem can be proved in the same way as above.
Theorem 5
if D = diag(a,b,c) then D.D = ( a2 , b2 , c2)
D.D.D = ( a3 , b3 , c3)
.....
This property can be generalised for D = diag(a,b,c,d,e,...,l).
Saturday, July 31, 2010
Algebra tricks
| Faster Multiplication of numbers | It is hard to believe till you see for yourself that you will be able to multiply giant 5 digit numbers in just one or 2 lines. For example : How fast can you calculate 4 3 2 6 1 x 3 4 8 2 3 = ? |
| Faster Algebra Multiplication ( from the basic to advanced level ) | Without any extra effort, you will be able to multiply Algebra polynomials ( all types ) directly . |
| Faster Algebra Division | Be prepared to be shocked !! As you would be able to do Long Division of Algebra in just one line and be able to find Quotient & Remainder both in just few seconds. |
| Superfast Way to solve the Algebra Equations ( all types ) | Effortlessly you would be able to solve any algebra equation ( from the simple linear equation to difficult quadratic equation or the most advanced Cubic Equations in few seconds ). |
| Easily solve Word Problems | Learn how to directly translate English into Mathematics. Solving long Word Problems was never this easy. |
Friday, June 11, 2010
Mathematical Formula Table and List
* Trygonometric Delights, and introduction of trygonometry free books
in this post I provide a simple review of the two free books that I found after browsing about trigonometry, there is no end to the discussion of the triangle, we can explore a part of this mathematics ranging from history, until a new theory about the true triangle, two…
* Best mathematics Integral help website
Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral.to learn more about integral, here i write some list…
* best online calculus help website
various online help of calculus available in internet, every those website have their advantage, we can choose which website that can fit with what we need.below some list calculus website with their unique facilities1. http://www.calculus-help.com/this website provide various facilities with unique multimedia tutorial, problem's this week can become interactive tutorial…
* Linear Methods of Applied Mathematics free book
Linear Methods of Applied MathematicsOrthogonal series, boundary-value problems, and integral operatorsEvans M. Harrell II and James V. Herod © Copyright 1994,1995,1996, 1997, 2000 by Evans M. HarrellII and James V. Herod. All rights reserved.This is a WWW textbook written by Evans M. Harrell II and James V. Herod, both of…
* Multivariable Calculus
This is a textbook for a course in multivariable calculus. It has been used for the past few years here at Georgia Tech. The notes are available as Adobe Acrobat documents,this book also available online at http://www.math.gatech.edu/~cain/notes/calculus.html, written by George Cain and James HerodTitle page and Table of Contents Table…
* Calculus from Gilbert Strang
his book is legal to download and available online at http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htmTOCW is pleased to make this textbook available online. Published in 1991 and still in print from Wellesley-Cambridge Press, the book is a useful resource for educators and self-learners alike. It is well organized, covers single variable and multivariable calculus…
in this post I provide a simple review of the two free books that I found after browsing about trigonometry, there is no end to the discussion of the triangle, we can explore a part of this mathematics ranging from history, until a new theory about the true triangle, two…
* Best mathematics Integral help website
Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral.to learn more about integral, here i write some list…
* best online calculus help website
various online help of calculus available in internet, every those website have their advantage, we can choose which website that can fit with what we need.below some list calculus website with their unique facilities1. http://www.calculus-help.com/this website provide various facilities with unique multimedia tutorial, problem's this week can become interactive tutorial…
* Linear Methods of Applied Mathematics free book
Linear Methods of Applied MathematicsOrthogonal series, boundary-value problems, and integral operatorsEvans M. Harrell II and James V. Herod © Copyright 1994,1995,1996, 1997, 2000 by Evans M. HarrellII and James V. Herod. All rights reserved.This is a WWW textbook written by Evans M. Harrell II and James V. Herod, both of…
* Multivariable Calculus
This is a textbook for a course in multivariable calculus. It has been used for the past few years here at Georgia Tech. The notes are available as Adobe Acrobat documents,this book also available online at http://www.math.gatech.edu/~cain/notes/calculus.html, written by George Cain and James HerodTitle page and Table of Contents Table…
* Calculus from Gilbert Strang
his book is legal to download and available online at http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htmTOCW is pleased to make this textbook available online. Published in 1991 and still in print from Wellesley-Cambridge Press, the book is a useful resource for educators and self-learners alike. It is well organized, covers single variable and multivariable calculus…
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