### Matrices

Definition :
The arrangement of elements in a square (or) in a rectangle is called a Matrix.

The rows are counted from top to bottom and the columns are counted from left to right.
There are 3 rows and 4 columns in the e.g  (i) and it is called a matrix of the type (or order)
3 × 4 (or)  3 × 4 matrix (to be read as 3 by 4 matrix).
The numbers 1, 2, 3, 4... are consistuting this matrix, are called the Elements of the martix.
Matrices are noticed by capital letters.
Matrices are enclosed in
(i)  square brackets [   ]

(ii) Parenthesis  (   )
(iii) Double bars
Note: (i): In matrix order, we always note row number first and column number next.

Type of Matrices

Square matrix :
If the matrix has equal number of rows and columns then the matrix is called a square martix.

Here aij denotes in general the element occuring in  ith  row  and  jth column.

Rectangular martix :
If the matrix has different number of rows and columns then the martix is called a Rectangular matrix.

Row Matrix :
If the matrix has only one row then the matrix is called a row matrix.

Column matrix :
If the matrix has only one column then it is called a column matrix.

Null matrix:
If every element of a matrix is ZERO then it is called a null (or) zero matrix. A null matrix of the type m × n is denoted by Om × n (or) in short by O.

Diagonal matrix :
If all the elements of × a square matrix are zero except those of principal diagonal then the matrix is called a Diagonal matrix. It is denoted by Diag.

Note (ii) :
In a square matrix all those elements aij for which i = j i.e. a11, a22, a33 ... are called the elements of the principal diagonal.
Note (iii) :
The diagonal from the first element of the first row to the last element of the last row is called the principal diagonal of the square matrix.
Note (iv):
The sum of the diagonal elements is called trace of the matrix. It is denoted by "Tr".

Scalar matrix :
A square matrix in which the diagonal elements are equal, all other elements being zeros, is called a scalar matrix.

Unit matrix :
A square matrix in which each diagonal element is unity, all other elements being zeros, is called a unit (or) an identity matrix. Unit matrix of order n is denoted by In.

Upper triangular matrix :
A square matrix A = [aij] is called an upper triangular matrix if aij  =  0    i > j.

In an upper triangular matrix, all elements below the principal diagonal are zeros.

Lower triangular matrix :
A square matrix B  =  [bij] is called a lower triangular matrix  if bij  =  0    i  <  j

In lower triangular matrix, all elements above the principal diagonal are zeros.
Note (v) : A triangular matrix A: [aij]n × n is called a strictly triangular if aii = 0   i  =  1, 2... n.

Nilpotent - matrix :
If An = 0 for n ∈ N then the square matrix A is called a Nilpotent matrix. If n is the least positive integer for which An = 0, then n is called the index of the nilpotent matrix A.

A is a nilpotant matrix of index 2.

Idempotent matrix :
If A2  =  A then A is called an idempotent matrix

A is an idempotent matrix

Involutory matrix :
If A2  =  I then the square matrix A is called an involutory matrix.

A is an involutory matrix

Inverse matrix :
If for a square matrix A there exists another matrix B such that AB = BA = I then B is called inverse of A.
In such case, A is said to be invertible and we write A-1 = B
AA-1  =   A-1 A  =  I

Transpose matrix :
The matrix obtained from any given matrix A by interchanging its rows and columns is called the transpose of the matrix A  and  is denoted by   AT (or) A'.
If  A  =  [aij]m × n then AT = [a'ji]n × m where  a'ji  =  aij

Symmetric matrix :
If AT =  A then the Square matrix A is called a symmetric matrix.
A = [aij]m × n  is Symmetric matrix  aij = aji  i, j

Minor :
Let A = [aij] be a Square matrix. The Minor of an element aij in A is the determinant of the square matrix that remains after deleting the ith  row  and jth  colum of A.

Properties of Determinant
(1) If rows and columns in a Square matrix A are interchanged then the value of its determinant remains unaltered.
i.e. detA  =  detAT
(2) If any two rows (or) columns are interchanged then the determinant of a Square matrix changes sign.

(3) If any two rows (or) columns are identical then the value of the determinant is zero.
(4) If all the elements of a row (or) column of a Square matrix are multiplied by a number k then the determinant of the resulting matrix is equal to k times the determinant of the original matrix.
(5) If each element in a row (or) column of a Square matrix is the sum of two terms then its determinat can be expressed as the sum of two determinants of two Square matrices of the same order.
(6) If the elements of a row (or) column of a Square matrix are added with k times the corresponding elements of any other row (or) column then the value of the determinant of the resulting matrix is unaltered.
(7) The sum of the product of the elements of any row (or) column of a Square matrix with the cofactors of the corresponding elements of any other row (or) column is zero.
i.e a1A2  +  b1B2  +  c1C2  =  a2A1 + b2B1  +  c2C1  =  a3A1  +  b3B1  +  c3C1  =  0

Submatrix :
A matrix attained by deleting some rows or columns (or both) of a matrix is called a submatrix.

Rank of a matrix :
Suppose A is a non-zero matrix. A positive integer 'r' is said to be the rank of A if
(i) there exists a non-zero r-rowed minor of A.
(ii) every (r+1)-rowed minor of A (if exists) is zero.
Note :  The rank of a zero matrix is zero.

Singular matrix :
If the determinant of a Square matrix is zero then it is called a singular matrix.

Note: Suppose A is a non-zero 3 × 3 matrix. If A is a singular matrix and every 2 × 2 Submatrix is also singular then the rank of A is 1.
Note: Suppose A is non-zero matrix of order 3 × 4 (or) 4 × 3. The rank of A is the maximum of ranks of all 3 × 3 submatrices of A.

Conceptual Theorems
1. If A is a Matrix of the type m × n and I is the unit matrix then show that AIn  = A =  ImA

3. Show that multiplication of matrices is associative i.e. A(BC)  =  (AB)C

4. If A is mxp matrix and B is p × n matrix then show that (AB)T = BTAT.

Posted Date : 06-11-2020

గమనిక : ప్రతిభ.ఈనాడు.నెట్‌లో కనిపించే వ్యాపార ప్రకటనలు వివిధ దేశాల్లోని వ్యాపారులు, సంస్థల నుంచి వస్తాయి. మరి కొన్ని ప్రకటనలు పాఠకుల అభిరుచి మేరకు కృత్రిమ మేధస్సు సాంకేతికత సాయంతో ప్రదర్శితమవుతుంటాయి. ఆ ప్రకటనల్లోని ఉత్పత్తులను లేదా సేవలను పాఠకులు స్వయంగా విచారించుకొని, జాగ్రత్తగా పరిశీలించి కొనుక్కోవాలి లేదా వినియోగించుకోవాలి. వాటి నాణ్యత లేదా లోపాలతో ఈనాడు యాజమాన్యానికి ఎలాంటి సంబంధం లేదు. ఈ విషయంలో ఉత్తర ప్రత్యుత్తరాలకు, ఈ-మెయిల్స్ కి, ఇంకా ఇతర రూపాల్లో సమాచార మార్పిడికి తావు లేదు. ఫిర్యాదులు స్వీకరించడం కుదరదు. పాఠకులు గమనించి, సహకరించాలని మనవి.

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