Elementary matrix already has now becomes as integral part of the mathematical background necessary in field of electrical / computer engineering/chemistry.
A matrix is any rectangular array of numbers written within braekets. A matrix is usually represented by a capital letter and classified by its dimensions. The dimension of the matrices are the number of rows and columns.
A `m xx n` matrix is usually written as
`A_(mxxn)=[(a_11,a_12, cdots ,a_(1n)),(a_21,a_22, cdots ,a_(2n)),(a_m1,a_m2, cdots ,a_(mn))]`
(where `a_0` represents any number which lies `i^(th)` row (from top) & `j^(th)` column form left)
(i) The matrix is not a number. It has got no numerical value.
(ii) The determinant of matrix `A_(m xx n)=|A_(m xx m)|=|(a_11, cdots cdots,a_(1m)), (cdots,cdots cdots, cdots),(a_11, cdots cdots,a_(1m))|`
Abbreviated as :
`A = [ a_(i,j)] 1 le i le m ; 1 le j le n, i` denotes the row and `j` denotes the column is called a matrix of order `m xx n`. The elements of a matrix may be real or complex numbers. If all the elements of a matrix are real, the matrix is called real matrix.
Special Type of Matrices :
`(A) text(Row Matrix :)`
`A=[a_11,a_12, cdots cdots , a_(1n)]` having one row. `(1 xx n)` matrix. (or row vectors)
`(B) text(Column Matrix :)`
`A=[(a_11),(a_21),(vdots),(a_(m1))]` having one column. `(m xx 1)` matrix (or column vectors).
`(C) text(Zero or Null Matrix :)`
`(A= O_(mxx n))` An `m xx n` matrix all whose entries are zero.
`A=[(0,0),(0,0),(0,0)]` is a `3 xx 2` null matrix & `B =[(0,0,0),(0,0,0),(0,0,0)]` is `3 xx 3` null matrix.
`(D) text(Horizontal Matrix :)`
A matrix of order `m xx n` is a horizontal matrix if `n > m` .
`[(1,2,3,4),(2,5,1,1)]`
`(E) text(Verical Matrix :)`
A matix of order `m xx n` is a vertical matrix if `m > n`.
`[(2,5),(1,1),(3,6),(2,4)]`
Note: Every row matrix is also a Horizontal but not the converse.
||ly every column matrix is also a vertical matrix but not the converse.
`(F) text(Square Matrix :)` (Order `n`)
If number of rows = number of column, matrix is a square matrix. A real square matrix all whose elements are positive is called a positive matrices. Such matrices have application in mechanics and economics.
i.e. `[(8,9,5),(2,3,4),(3,-2,5)]`
`(G) text( Rectangular Matrix :)`
A matrix is said to be rectangular matrix, if the number of rows and the number of columns are not equal i.e., a matrix `A= [a_(ij)]_(mxxn)` is called a rectangular matrix, if `m != n.`
e.g. `A = [(1,5,3,2),(9,6,-6,4),(1,8,9,3)]`
Note :
(i) In a square matrix the pair of elements `a_(ij)` & `a_(ji)` are called Conjugate Elements.
e.g. in the matrix `((a_11, a_12),(a_21,a_22))` , `a_21` and `a_12` are conjugate elements.
(ii) The elements `a_11 , a_22 , a_33 , ...... a_(nn)` are called Diagonal Elements . The line along which the diagonal elements lie is called "Principal or Leading" diagonal.
The quantity `sum a_(ij) =` trace of the matrix written as, `( t_r)A= t_r(A)`
Note :
(i) Minimum number of zeros in an upper or lower triangular matrix of order `n`.
`=1+2+3+.............+(n-1)=(n(-1))/2`
(ii) Minimum number of cyphers in a diagonal/scalar/unit matrix of order `n = n(n - 1)` and maximum number of cyphers `= n^2 - 1`.
"It is to be noted that with every square matrix there is a corresponding determinant formed by the elements of `A` in the same order." If `|A| = 0` then `A` is called a singular matrix and if `|A| ne 0` then a is called a non singular matrix.
Note :
If `A=[(0,0),(0,0)]` then det. `A=0` but not conversely.
`(H)` `text(Vertical Matrix)`
A matrix is said to be vertical matrix, if the number of rows is greater than
the number of columns i.e., a matrix `A= [a_(ij)]_(m xx n)` is said to vertical matrix, iff
`m > n `
For example, `A = [(2,3,4),(0,-1,7),(3,4,4),(2,7,9),(-1,2,-5)]_(5 xx 3)` is a vertical matrix. [ ∵ number of rows (5) >number of columns (3)]
`(I)` `text(Sub-Matrix)`
A matrix which is obtained from a given matrix by deleting any number of
rows and number of columns is called a sub-matrix of the given matrix.
For example,
`[(3,4),(-2,5)]` is a sub-matrix of `[(8,9,5),(2,3,4),(3,-2,5)]`
{except : Identity, Triangular, Diagonal, Scalar, Symmetric, Skew-Symmetric, Nilpotent, Idempotent, Hermitian, Skew-Hermitian}
Elementary matrix already has now becomes as integral part of the mathematical background necessary in field of electrical / computer engineering/chemistry.
A matrix is any rectangular array of numbers written within braekets. A matrix is usually represented by a capital letter and classified by its dimensions. The dimension of the matrices are the number of rows and columns.
A `m xx n` matrix is usually written as
`A_(mxxn)=[(a_11,a_12, cdots ,a_(1n)),(a_21,a_22, cdots ,a_(2n)),(a_m1,a_m2, cdots ,a_(mn))]`
(where `a_0` represents any number which lies `i^(th)` row (from top) & `j^(th)` column form left)
(i) The matrix is not a number. It has got no numerical value.
(ii) The determinant of matrix `A_(m xx n)=|A_(m xx m)|=|(a_11, cdots cdots,a_(1m)), (cdots,cdots cdots, cdots),(a_11, cdots cdots,a_(1m))|`
Abbreviated as :
`A = [ a_(i,j)] 1 le i le m ; 1 le j le n, i` denotes the row and `j` denotes the column is called a matrix of order `m xx n`. The elements of a matrix may be real or complex numbers. If all the elements of a matrix are real, the matrix is called real matrix.
Special Type of Matrices :
`(A) text(Row Matrix :)`
`A=[a_11,a_12, cdots cdots , a_(1n)]` having one row. `(1 xx n)` matrix. (or row vectors)
`(B) text(Column Matrix :)`
`A=[(a_11),(a_21),(vdots),(a_(m1))]` having one column. `(m xx 1)` matrix (or column vectors).
`(C) text(Zero or Null Matrix :)`
`(A= O_(mxx n))` An `m xx n` matrix all whose entries are zero.
`A=[(0,0),(0,0),(0,0)]` is a `3 xx 2` null matrix & `B =[(0,0,0),(0,0,0),(0,0,0)]` is `3 xx 3` null matrix.
`(D) text(Horizontal Matrix :)`
A matrix of order `m xx n` is a horizontal matrix if `n > m` .
`[(1,2,3,4),(2,5,1,1)]`
`(E) text(Verical Matrix :)`
A matix of order `m xx n` is a vertical matrix if `m > n`.
`[(2,5),(1,1),(3,6),(2,4)]`
Note: Every row matrix is also a Horizontal but not the converse.
||ly every column matrix is also a vertical matrix but not the converse.
`(F) text(Square Matrix :)` (Order `n`)
If number of rows = number of column, matrix is a square matrix. A real square matrix all whose elements are positive is called a positive matrices. Such matrices have application in mechanics and economics.
i.e. `[(8,9,5),(2,3,4),(3,-2,5)]`
`(G) text( Rectangular Matrix :)`
A matrix is said to be rectangular matrix, if the number of rows and the number of columns are not equal i.e., a matrix `A= [a_(ij)]_(mxxn)` is called a rectangular matrix, if `m != n.`
e.g. `A = [(1,5,3,2),(9,6,-6,4),(1,8,9,3)]`
Note :
(i) In a square matrix the pair of elements `a_(ij)` & `a_(ji)` are called Conjugate Elements.
e.g. in the matrix `((a_11, a_12),(a_21,a_22))` , `a_21` and `a_12` are conjugate elements.
(ii) The elements `a_11 , a_22 , a_33 , ...... a_(nn)` are called Diagonal Elements . The line along which the diagonal elements lie is called "Principal or Leading" diagonal.
The quantity `sum a_(ij) =` trace of the matrix written as, `( t_r)A= t_r(A)`
Note :
(i) Minimum number of zeros in an upper or lower triangular matrix of order `n`.
`=1+2+3+.............+(n-1)=(n(-1))/2`
(ii) Minimum number of cyphers in a diagonal/scalar/unit matrix of order `n = n(n - 1)` and maximum number of cyphers `= n^2 - 1`.
"It is to be noted that with every square matrix there is a corresponding determinant formed by the elements of `A` in the same order." If `|A| = 0` then `A` is called a singular matrix and if `|A| ne 0` then a is called a non singular matrix.
Note :
If `A=[(0,0),(0,0)]` then det. `A=0` but not conversely.
`(H)` `text(Vertical Matrix)`
A matrix is said to be vertical matrix, if the number of rows is greater than
the number of columns i.e., a matrix `A= [a_(ij)]_(m xx n)` is said to vertical matrix, iff
`m > n `
For example, `A = [(2,3,4),(0,-1,7),(3,4,4),(2,7,9),(-1,2,-5)]_(5 xx 3)` is a vertical matrix. [ ∵ number of rows (5) >number of columns (3)]
`(I)` `text(Sub-Matrix)`
A matrix which is obtained from a given matrix by deleting any number of
rows and number of columns is called a sub-matrix of the given matrix.
For example,
`[(3,4),(-2,5)]` is a sub-matrix of `[(8,9,5),(2,3,4),(3,-2,5)]`
{except : Identity, Triangular, Diagonal, Scalar, Symmetric, Skew-Symmetric, Nilpotent, Idempotent, Hermitian, Skew-Hermitian}